text: 1 Network Resource Sharing Games with Instantaneous Reciprocity Sofonias Hailu1, Ragnar Freij-Hollanti1, Alexis A. Dowhuszko2, and Olav Tirkkonen1 1Department of Communications and Networking, Aalto University, P.O. Box 13000, FI-00076 Aalto, Finland 2Centre Tecnològic de les Telecomunicacions de Catalunya (CTTC), Barcelona, Spain E-mail: {sofonias.hailu, ragnar.freij, olav.tirkkonen}@aalto.ﬁ, alexis.dowhuszko@cttc.es Abstract We propose a generic strategic network resource sharing game between a set of players representing operators. The players negotiate which sets of players share given resources, serving users with varying sensitivity to interference. We prove that the proposed game has a Nash equilibrium, to which a greedily played game converges. Furthermore, simulation results show that, when applied to inter- operator spectrum sharing in small-cell indoor oﬃce environment, the convergence is fast and there is a signiﬁcant performance improvement for the operators when compared to the default resource usage conﬁguration. Index Terms Game theory, N-person game, network resource sharing, inter-operator spectrum sharing I. Introduction Game theory has been widely used in analyzing and designing wireless network protocols. Often, game theoretical principles have been used as guiding light when striving for distributed solutions of NP-hard optimization problems. The idea is that if in a given networking situation, a

text: Network

text: Resource

text: Sharing

text: Games

text: with

text: Instantaneous

text: Reciprocity

text: Sofonias

text: Ragnar

text: Alexis

text: A.

text: and

text: Olav

text: of

text: Communications

text: and

text: Networking,

text: Aalto

text: University,

text: P.O.

text: Box

text: Aalto,

text: Finland

text: Tecnològic

text: de

text: les

text: Telecomunicacions

text: de

text: Catalunya

text: (CTTC),

text: Barcelona,

text: Spain

text: E-mail:

text: {sofonias.hailu,

text: ragnar.freij,

text: olav.tirkkonen}@aalto.ﬁ,

text: alexis.dowhuszko@cttc.es

text: Abstract

text: We

text: propose

text: a

text: generic

text: strategic

text: network

text: resource

text: sharing

text: game

text: between

text: a

text: set

text: of

text: players

text: representing

text: operators.

text: The

text: players

text: negotiate

text: which

text: sets

text: of

text: players

text: share

text: given

text: resources,

text: serving

text: users

text: with

text: varying

text: sensitivity

text: to

text: interference.

text: We

text: prove

text: that

text: the

text: proposed

text: game

text: has

text: a

text: Nash

text: equilibrium,

text: to

text: which

text: a

text: greedily

text: played

text: game

text: converges.

text: Furthermore,

text: simulation

text: results

text: show

text: that,

text: when

text: applied

text: to

text: inter-

text: operator

text: spectrum

text: sharing

text: in

text: small-cell

text: indoor

text: oﬃce

text: environment,

text: the

text: convergence

text: is

text: fast

text: and

text: there

text: is

text: a

text: signiﬁcant

text: performance

text: improvement

text: for

text: the

text: operators

text: when

text: compared

text: to

text: the

text: default

text: resource

text: usage

text: conﬁguration.

text: Index

text: Terms

text: Game

text: theory,

text: N-person

text: game,

text: network

text: resource

text: sharing,

text: inter-operator

text: spectrum

text: sharing

text: I.

text: Introduction

text: Game

text: theory

text: has

text: been

text: widely

text: used

text: in

text: analyzing

text: and

text: designing

text: wireless

text: network

text: protocols.

text: Often,

text: game

text: theoretical

text: principles

text: have

text: been

text: used

text: as

text: guiding

text: light

text: when

text: striving

text: for

text: distributed

text: solutions

text: of

text: NP-hard

text: optimization

text: problems.

text: The

text: idea

text: is

text: that

text: if

text: in

text: a

text: given

text: networking

text: situation,

text: a


text: 11 ∃ | { } of the players is selected, i.e. n a = Υ( ai ∈N), leading to the maximum possible payoﬀ n n n improvement for the player. Thus, there is no reason for the player to change the agreed resource utilization pattern. Lemma 2: The sequential 2-person game converges in one iteration. The strategy of the players is dominant. At least one of the players has a strongly dominant strategy. N { } Proof: Let = 1, 2 , and let a (cid:44) a . 1 2 − ≥ { } Case 1: Assume a N b0N 0 for both n = 1, 2. We have Υ( ai ∈N) = a if a N < a N. n n n 1 1 2 (cid:48) Thus, player 1 gets the maximum payoﬀ and the game converges. Moreover, Φ (a) > Φ (a ) 1 1 (cid:48) ∈ A ≥ (cid:48) (cid:48) ∈ A for all a and Φ (a) Φ (a ) for all a . The inequality in the latter case is tight 1 2 2 2 ∈ A | for a a N > a N. Thus, player 1 has a strongly dominant strategy while the strategy for 2 2 1 player 2 is weakly dominant. − ≤ { } Case 2: Assume a N b0N 0 for both n = 1, 2. We have Υ( ai ∈N) = a if a N > a N. n n n 1 1 2 (cid:48) Thus, player 1 gets the maximum payoﬀ and the game converges. Similarly, Φ (a) > Φ (a ) for 1 1 (cid:48) ∈ A ≥ (cid:48) (cid:48) ∈ A all a and Φ (a) Φ (a ) for all a . The inequality in the latter case is tight for 1 2 2 2 ∈ A | a a N < a N. Therefore, player 1 has a strongly dominant strategy while the strategy for 2 2 1 player 2 is weakly dominant. If a = a , both players get a maximum payoﬀ. The game converges. The strategy of both 1 2 players is strongly dominant. ≥ For N 3, we propose an alternative game, with faster convergence, although towards a ⊆ N suboptimal point. Here, the game is played between the players in a subset S , and the ∈ S dimension is reduced by only negotiating the parameters b{ }, n and bS. Now, the strategy n of a player becomes a = arg max g (a) n n a (11) ∈ A subject to a S n where A { ∈ A | T ∈ P T S |T | } S = a aT = b0T for all , (cid:44) , > 1 . (12) n n n However, the players need to have an a priori agreed rule on how to choose the sequence of the subsets. With this approach, the players may play a sequence of single dimensional games as summarized in Algorithm 2. Theorem 4: The sequential single dimensional N-person game converges.

label: paragraph

font: GKOZSN+txsy

text: of

text: the

text: players

text: is

text: selected,

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text: n

text: a

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text: Υ(

text: ai

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text: the

text: maximum

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text: payoﬀ

text: n

text: n

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text: improvement

text: for

text: the

text: player.

text: Thus,

text: there

text: is

text: no

text: reason

text: for

text: the

text: player

text: to

text: change

text: the

text: agreed

text: resource

text: utilization

text: pattern.

text: Lemma

text: The

text: sequential

text: game

text: converges

text: in

text: one

text: iteration.

text: The

text: strategy

text: of

text: the

text: players

text: is

text: dominant.

text: At

text: least

text: one

text: of

text: the

text: players

text: has

text: a

text: strongly

text: dominant

text: strategy.

text: N

text: Proof:

text: Let

text: and

text: let

text: a

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text: Case

text: Assume

text: a

text: N

text: for

text: both

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text: a

text: if

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text: N.

text: n

text: n

text: n

text: Thus,

text: player

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text: the

text: maximum

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text: and

text: the

text: game

text: converges.

text: Moreover,

text: Φ

text: (a)

text: Φ

text: (a

text: A

text: A

text: for

text: all

text: a

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text: Φ

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text: Φ

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text: a

text: The

text: inequality

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text: the

text: latter

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text: A

text: for

text: a

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text: N.

text: Thus,

text: player

text: has

text: a

text: strongly

text: dominant

text: strategy

text: while

text: the

text: strategy

text: for

text: player

text: is

text: weakly

text: dominant.

text: Case

text: Assume

text: a

text: N

text: for

text: both

text: n

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text: have

text: Υ(

text: ai

text: ∈N)

text: a

text: if

text: a

text: N

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text: N.

text: n

text: n

text: n

text: Thus,

text: player

text: gets

text: the

text: maximum

text: payoﬀ

text: and

text: the

text: game

text: converges.

text: Similarly,

text: Φ

text: (a)

text: Φ

text: (a

text: for

text: A

text: A

text: all

text: a

text: and

text: Φ

text: (a)

text: Φ

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text: for

text: all

text: a

text: The

text: inequality

text: in

text: the

text: latter

text: case

text: is

text: tight

text: for

text: A

text: a

text: a

text: N

text: a

text: N.

text: Therefore,

text: player

text: has

text: a

text: strongly

text: dominant

text: strategy

text: while

text: the

text: strategy

text: for

text: player

text: is

text: weakly

text: dominant.

text: If

text: a

text: a

text: both

text: players

text: get

text: a

text: maximum

text: payoﬀ.

text: The

text: game

text: converges.

text: The

text: strategy

text: of

text: both

text: players

text: is

text: strongly

text: dominant.

text: For

text: N

text: we

text: propose

text: an

text: alternative

text: game,

text: with

text: faster

text: convergence,

text: although

text: towards

text: a

text: N

text: suboptimal

text: point.

text: Here,

text: the

text: game

text: is

text: played

text: between

text: the

text: players

text: in

text: a

text: subset

text: S

text: and

text: the

text: S

text: dimension

text: is

text: reduced

text: by

text: only

text: negotiating

text: the

text: parameters

text: b{

text: n

text: and

text: bS.

text: Now,

text: the

text: strategy

text: n

text: of

text: a

text: player

text: becomes

text: a

text: arg

text: max

text: g

text: (a)

text: n

text: n

text: a

text: A

text: subject

text: to

text: a

text: S

text: n

text: where

text: A

text: A

text: T

text: P

text: T

text: S

text: |T

text: S

text: a

text: aT

text: for

text: all

text: n

text: n

text: n

text: However,

text: the

text: players

text: need

text: to

text: have

text: an

text: a

text: priori

text: agreed

text: rule

text: on

text: how

text: to

text: choose

text: the

text: sequence

text: of

text: the

text: subsets.

text: With

text: this

text: approach,

text: the

text: players

text: may

text: play

text: a

text: sequence

text: of

text: single

text: dimensional

text: games

text: as

text: summarized

text: in

text: Algorithm

text: Theorem

text: The

text: sequential

text: single

text: dimensional

text: N-person

text: game

text: converges.


text: 19 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Default 0.2 MRG 0.1 CS-SR CS-LR 0 0 10 20 30 40 50 60 70 80 User rates [Mbps] Fig. 6: Four-players MRG. Visiting probability = 0. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Default 0.2 RPG 0.1 CS-SR CS-LR 0 0 10 20 30 40 50 60 User rates [Mbps] Fig. 7: Four-players RPG. Visiting probability = 0.5. maximum utility improvement. The same baseline and upper bounds as the two-players game are also used for the four-players games. The results for MRG with visiting probability of 0 and RPG with visiting probability of 0.5 have a signiﬁcant gain comparing to the results with ’default’ utilization pattern. However, comparison with the upper bounds indicates that there is still room for improvement. On the other hand, Figures 8 and 9 show the number of iteration until the four-players MRG and RPG converge. The convergenceforthe sequential game summarized in Algorithm 1 is referred as ’MDSG’ and the one in in Algorithm 2 as ’SDSG’. The results indicate that the FDC FDC ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Default

text: MRG

text: CS-SR

text: CS-LR

text: User

text: rates

text: [Mbps]

text: Fig.

text: Four-players

label: caption

text: MRG.

text: Visiting

text: probability

text: Default

text: RPG

text: CS-SR

text: CS-LR

text: User

text: rates

text: [Mbps]

text: Fig.

text: Four-players

text: RPG.

text: Visiting

text: probability

text: maximum

text: utility

text: improvement.

text: The

text: same

text: baseline

text: and

text: upper

text: bounds

text: as

text: the

text: two-players

text: game

text: are

text: also

text: used

text: for

text: the

text: four-players

text: games.

text: The

text: results

text: for

text: MRG

text: with

text: visiting

text: probability

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text: and

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text: visiting

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text: have

text: a

text: signiﬁcant

text: gain

text: comparing

text: to

text: the

text: results

text: with

text: ’default’

text: utilization

text: pattern.

text: However,

text: comparison

text: with

text: the

text: upper

text: bounds

text: indicates

text: that

text: there

text: is

text: still

text: room

text: for

text: improvement.

text: On

text: the

text: other

text: hand,

text: Figures

text: and

text: show

text: the

text: number

text: of

text: iteration

text: until

text: the

text: four-players

text: MRG

text: and

text: RPG

text: converge.

text: The

text: convergenceforthe

text: sequential

text: game

text: summarized

text: in

text: Algorithm

text: is

text: referred

text: as

text: ’MDSG’

text: and

text: the

text: one

text: in

text: in

text: Algorithm

text: as

text: ’SDSG’.

text: The

text: results

text: indicate

text: that

text: the

text: FDC

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text: in

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text: GLOBECOM,

text: Dec.

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text: DaSilva,

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text: in

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text: IEEE

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text: pp.

text: I.

text: Bistritz

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text: Leshem,

text: “Asymptotically

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text: a

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text: in

text: Proc.

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text: pp.

text: J.

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text: J.

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text: 7 patterns, but only in the default sharing pattern, and hence in the resolution rule, which is deﬁned in Section . B. Strategy space A A Let be the strategy space of player n, meaning the set of allowed bids. We deﬁne as n (cid:88) A ⊂ { ∪ ∗} the subspace R 2N given by the constraints n a 1 | S| = , S N ∈P S n ∗ ∀ ⊂ N (8) a S = n (cid:60) S, S , n ≥ ∀ ∈ ⊂ N a S 0 n S, S n which is a compact convex set. This means that a has to respect the instantaneous reciprocity n ∗ constraint (6). Moreover, if n (cid:60) S , then we require a to take the void value , which should nS be interpreted as not having any prefered value of b . This is natural to require, to not allow S players to obstruct an agreement that does not aﬀect her payoﬀ, between a set of players in which she is not a member. C. An a-priori resolution rule The game has an a-priori agreed rule to resolve the strategies of the players. Denote the resolution rule by ∪ ∗ · → Υ : (R )N 2N R2N, { } ∈ mapping the joint strategy of the players a ∈N to the resource allocation b R2N. The resolution n n ∈ F rule Υ will also depend on a default resource distribution b0 , as described in Section . We also propose a sequential game, where the outcome of one round will be the default distribution ∈ F of the next. This is only one of the reasons why we allow arbitrary b0 . (cid:89) (cid:89) ∈ ∪ ∗ For a bid a (R )2N, deﬁne the set n × I = [min(a , b0 ), max(a , b0 )] R. n nS S nS S ∈P P S S(cid:60) { ∈n } n Requiring the outcome b : n S to remain within I , we guarantee that no player has to S n change her default usage pattern more than she is willing(cid:88)to according to her bid. We propose the following resolution rule: { } | − | Υ( a ∈N) = argmax b b0 n n S S b ⊂N S ∈ F (9) subject to b ∈ ∈ N b I for all n . n ##LTLine## ##LTLine##

text: patterns,

text: but

text: only

text: in

text: the

text: default

text: sharing

text: pattern,

text: and

text: hence

text: in

text: the

text: resolution

text: rule,

text: which

text: is

text: deﬁned

text: in

text: Section

text: B.

label: section

text: Strategy

text: space

text: A

text: A

text: Let

text: be

text: the

text: strategy

text: space

text: of

text: player

text: n,

text: meaning

text: the

text: set

text: of

text: allowed

text: bids.

text: We

text: deﬁne

text: as

text: n

text: A

text: the

text: subspace

text: R

font: XMRDOS+txsyb

text: given

text: by

text: the

text: constraints

text: n

text: a

text: S|

font: CNUNNO+rtxmi

text: S

text: N

text: ∈P

text: S

text: n

text: N

text: a

text: S

text: n

text: S,

text: S

text: n

text: N

text: a

text: S

text: n

text: S,

text: S

text: n

text: which

text: is

text: a

text: compact

text: convex

text: set.

text: This

text: means

text: that

text: a

text: has

text: to

text: respect

text: the

text: instantaneous

text: reciprocity

text: n

text: constraint

text: Moreover,

text: if

text: n

text: S

text: then

text: we

text: require

text: a

text: to

text: take

text: the

text: void

text: value

text: which

text: should

text: nS

text: be

text: interpreted

text: as

text: not

text: having

text: any

text: prefered

text: value

text: of

text: b

text: This

text: is

text: natural

text: to

text: require,

text: to

text: not

text: allow

text: S

text: players

text: to

text: obstruct

text: an

text: agreement

text: that

text: does

text: not

text: aﬀect

text: her

text: payoﬀ,

text: between

text: a

text: set

text: of

text: players

text: in

text: which

text: she

text: is

text: not

text: a

text: member.

text: C.

text: An

text: a-priori

text: resolution

text: rule

text: The

text: game

text: has

text: an

text: a-priori

text: agreed

text: rule

text: to

text: resolve

text: the

text: strategies

text: of

text: the

text: players.

text: Denote

text: the

text: resolution

text: rule

text: by

text: Υ

text: (R

text: )N

text: mapping

text: the

text: joint

text: strategy

text: of

text: the

text: players

text: a

text: ∈N

text: to

text: the

text: resource

text: allocation

text: b

text: The

text: resolution

text: n

text: n

text: F

text: rule

text: Υ

text: will

text: also

text: depend

text: on

text: a

text: default

text: resource

text: distribution

text: as

text: described

text: in

text: Section

text: We

text: also

text: propose

text: a

text: sequential

text: game,

text: where

text: the

text: outcome

text: of

text: one

text: round

text: will

text: be

text: the

text: default

text: distribution

text: F

text: of

text: the

text: next.

text: This

text: is

text: only

text: one

text: of

text: the

text: reasons

text: why

text: we

text: allow

text: arbitrary

text: For

text: a

text: bid

text: a

text: (R

text: deﬁne

text: the

text: set

text: n

text: I

text: [min(a

text: max(a

text: R.

text: n

text: nS

text: S

text: nS

text: S

text: ∈P

text: P

text: S

text: ∈n

text: n

text: Requiring

text: the

text: outcome

text: b

text: n

text: S

text: to

text: remain

text: within

text: I

text: we

text: guarantee

text: that

text: no

text: player

text: has

text: to

text: S

text: n

text: change

text: her

text: default

text: usage

text: pattern

text: more

text: than

text: she

text: is

text: according

text: to

text: her

text: bid.

text: We

text: propose

text: the

text: following

text: resolution

text: rule:

text: Υ(

text: a

text: ∈N)

text: argmax

text: b

text: n

text: n

text: S

text: S

text: b

text: ⊂N

text: S

text: F

text: subject

text: to

text: b

text: N

text: b

text: I

text: for

text: all

text: n

text: n

text: ##LTLine##

text: ##LTLine##


text: 8 Observe that, while in general the maximum might not be unique, it will be almost surely, assuming the bids are drawn from a continuous probability distribution. This will in turn happen if the players play optimally, and the players’ utility functions g are generic enough. We may n also observe that, while it is in general computationally unfeasible to ﬁnd the largest vector in a given set, the resolution rule can be evaluated eﬃciently, thanks to the following lemma. Theorem 1: The resolution rule Υ is the solution of a linear optimization problem. Proof: The proof is easy but technical, and is postponed to Appendix A. IV. Existence of equilibrium point We will begin by stating and proving a standard game theoretic lemma in a version that suits our settings. For game-theoretic terminology, we refer to [18], and for topological notions we refer to [19]. N A Lemma 1: Let ( , , Φ) be an N-player game that satisﬁes the following three criteria: 1) The payoﬀ function Φ is upper semicontinuous. A 2) The joint strategy space is convex and compact. ∈ N { } 3) For each player n , and all joint strategies a− = a : i (cid:44) n , the set n iS γ (a− ) = argmax Φ(a) n n a n is convex. N A Then ( , , Φ) has a Nash equilibrium point. (cid:81) A A ∇ A → Proof: Let P( ) be the set of all subspaces of . Consider the set valued map : A ∇ P( ), given by (a) = γ (a− ). By upper semicontinuity of Φ, γ is closed, and by assumption n n n n ∇ (3) it is also convex. As (a) is a product of ﬁnitely many compact convex sets, it is itself compact ∇ and convex. It also follows from upper semicontinuity of Φ that the is upper hemicontinuous. → Now by Kakutani’s theorem [20], any function X P(X) on a compact domain X, that is (cid:81) upper-hemicontinuous and takes closed and convex values, has a ﬁxed point. Hence, there exists ∈ A ∈ ∇ ∈ a such that a (a) = γ (a− ). This means that a γ (a− ) for every n, which in n n n n n n turn means that no player can improve her outcome by changing her bid. Hence, a is a Nash equilibrium. We now show that 1 applies to the game(cid:89)described in Section . Theorem 2: The game N A A ◦ ( , = , Φ = g Υ) n ∈N n

text: Observe

text: that,

text: while

text: in

text: general

text: the

text: maximum

text: might

text: not

text: be

text: unique,

text: it

text: will

text: be

text: almost

text: surely,

text: assuming

text: the

text: bids

text: are

text: drawn

text: from

text: a

text: continuous

text: probability

text: distribution.

text: This

text: will

text: in

text: turn

text: happen

text: if

text: the

text: players

text: play

text: optimally,

text: and

text: the

text: players’

text: utility

text: functions

text: g

text: are

text: generic

text: enough.

text: We

text: may

text: n

text: also

text: observe

text: that,

text: while

text: it

text: is

text: in

text: general

text: computationally

text: unfeasible

text: to

text: ﬁnd

text: the

text: largest

text: vector

text: in

text: a

text: given

text: set,

text: the

text: resolution

text: rule

text: can

text: be

text: evaluated

text: eﬃciently,

text: thanks

text: to

text: the

text: following

text: lemma.

text: Theorem

text: The

text: resolution

text: rule

text: Υ

text: is

text: the

text: solution

text: of

text: a

text: linear

text: optimization

text: problem.

text: Proof:

text: The

text: proof

text: is

text: easy

text: but

text: technical,

text: and

text: is

text: postponed

text: to

text: Appendix

text: A.

text: IV.

text: Existence

text: of

text: equilibrium

text: point

text: We

text: will

text: begin

text: by

text: stating

text: and

text: proving

text: a

text: standard

text: game

text: theoretic

text: lemma

text: in

text: a

text: version

text: that

text: suits

text: our

text: settings.

text: For

text: game-theoretic

text: terminology,

text: we

text: refer

text: to

text: and

text: for

text: topological

text: notions

text: we

text: refer

text: to

text: N

text: A

text: Lemma

text: Let

text: Φ)

text: be

text: an

text: N-player

text: game

text: that

text: satisﬁes

text: the

text: following

text: three

text: criteria:

text: The

text: payoﬀ

text: function

text: Φ

text: is

text: upper

text: semicontinuous.

text: A

text: The

text: joint

text: strategy

text: space

text: is

text: convex

text: and

text: compact.

text: N

text: For

text: each

text: player

text: n

text: and

text: all

text: joint

text: strategies

text: a−

text: a

text: i

text: n

text: the

text: set

text: n

text: iS

text: γ

text: (a−

text: argmax

text: Φ(a)

text: n

text: n

text: a

text: n

text: is

text: convex.

text: N

text: A

text: Then

text: Φ)

text: has

text: a

text: Nash

text: equilibrium

text: point.

text: A

text: A

text: A

text: Proof:

text: Let

text: P(

text: be

text: the

text: set

text: of

text: all

text: subspaces

text: of

text: Consider

text: the

text: set

text: valued

text: map

text: A

text: P(

text: given

text: by

text: (a)

text: γ

text: (a−

text: By

text: upper

text: semicontinuity

text: of

text: Φ,

text: γ

text: is

text: closed,

text: and

text: by

text: assumption

text: n

text: n

text: n

text: n

text: it

text: is

text: also

text: convex.

text: As

text: (a)

text: is

text: a

text: product

text: of

text: ﬁnitely

text: many

text: compact

text: convex

text: sets,

text: it

text: is

text: itself

text: compact

text: and

text: convex.

text: It

text: also

text: follows

text: from

text: upper

text: semicontinuity

text: of

text: Φ

text: that

text: the

text: is

text: upper

text: hemicontinuous.

text: Now

text: by

text: Kakutani’s

text: theorem

text: any

text: function

text: X

text: P(X)

text: on

text: a

text: compact

text: domain

text: X,

text: that

text: is

text: upper-hemicontinuous

text: and

text: takes

text: closed

text: and

text: convex

text: values,

text: has

text: a

text: ﬁxed

text: point.

text: Hence,

text: there

text: exists

text: A

text: a

text: such

text: that

text: a

text: (a)

text: γ

text: (a−

text: This

text: means

text: that

text: a

text: γ

text: (a−

text: for

text: every

text: n,

text: which

text: in

text: n

text: n

text: n

text: n

text: n

text: n

text: turn

text: means

text: that

text: no

text: player

text: can

text: improve

text: her

text: outcome

text: by

text: changing

text: her

text: bid.

text: Hence,

text: a

text: is

text: a

text: Nash

text: equilibrium.

text: We

text: now

text: show

text: that

text: applies

text: to

text: the

text: in

text: Section

text: Theorem

text: The

text: game

text: N

text: A

text: A

text: Φ

text: g

text: Υ)

text: n

text: ∈N

text: n


text: No. X Chinese Physics C Vol. xx, No. x (201x) xxxxxx 12 Fig. 6. (color online) Comparison of distributions between data and MC for the number of charged tracks at (a) 2.2324 GeV, (b) 2.4000 GeV, (c) 2.8000 GeV, (d) 3.0500 GeV, (e) 3.0600 GeV, (f) 3.0800 GeV, (g) 3.400 GeV, (h) 3.500 GeV, (i) 3.5424 GeV, (j) 3.5538 GeV, (k) 3.5611 GeV, (l) 3.6002 GeV, (m) 3.6500 GeV, (n) 3.6710 GeV. The dots denote data, and the open bars denote MC. 010201-12 ##LTLine##

text: No.

text: X

text: Chinese

text: Physics

text: C

text: Vol.

text: xx,

text: No.

text: x

text: xxxxxx

text: Fig.

text: (color

text: online)

text: Comparison

text: of

text: distributions

text: between

text: data

text: and

text: MC

text: for

text: the

text: number

text: of

text: charged

text: tracks

text: at

text: (a)

text: GeV,

text: (b)

text: GeV,

text: (c)

text: GeV,

text: (d)

text: GeV,

text: (e)

text: GeV,

text: (f)

text: GeV,

text: (g)

text: GeV,

text: (h)

text: GeV,

text: (i)

text: GeV,

text: (j)

text: GeV,

text: (k)

text: GeV,

text: (l)

text: GeV,

text: (m)

text: GeV,

text: (n)

text: GeV.

text: The

text: dots

text: denote

text: data,

text: and

text: the

text: open

text: bars

text: denote

text: MC.

text: ##LTLine##


text: No. X Chinese Physics C Vol. xx, No. x (201x) xxxxxx 6 by ﬁtting the cross section mode by mode. These empirical functions are used in the generator for the calculation of the ISR correction factor and event type sampling. The angular distribution for ISR photons is implemented according to Eq. (6). However, − angular distributions are implemented only for two-body decays, namely, 1 cos2 θ for PP (where P is a pseudoscalar meson) modes, and 1+αcos2 θ for the PV (α = 1) and BB¯ modes, where V ¯ is a vector meson, and B is a baryon. The angular distribution parameter α for the BB mode is taken as the quark model prediction [43]. The phase space model is used for multi-body decays. 2.4 LUND Area Law model − The hadronic events produced in the e+e annihilation are evolved as follows. As the ﬁrst − step, a quark-antiquark (qq¯) pair is produced from a virtual photon, coupled to the e+e pair. Then the qq¯ branching proceeds via emitting gluons, and further develops into hadrons. In the high energy region, the cluster model (e.g. HERWIG [44]) and LUND string model (e.g. JETSET/PYTHIA[45]) are available and precise enough to describe the hadronic fragmentation with parameters optimized at boson Z peak. However, in the intermediate and low energy region, parameters need to be optimized or a new model is desirable to describe the light quark fragmentation. In the tau-charm energy region, the LUARLW model [6] has been proposed to estimate the multiplicity distribution for primary hadrons produced from the string fragmentation. The probability distribution reads: µn − − P = exp[c +c (n µ)+c (n µ)2], (8) n 0 1 2 n! √ with µ = α + β exp(γ s), where c ,c ,c ,α,β and γ are parameters to be tuned with data. 0 1 2 An interface to access the LUARLW model is designed in the BesEvtGen [5] framework, and is only used to generate the primary hadrons. The further decays into light hadrons are realized with BesEvtGen [5]. 2.5 Monte Carlo algorithm The event sampling proceeds via two steps. Firstly, the mass of the hadron system, M , hadrons √ is sampled according to the distribution of the observed cross section, i.e. dσ(s)/dm, for the − → − process e+e γ X according to Eq. (1). For simplicity, the ISR energy, s M , is ISR i hadrons imposed on a single photon. The second step is to sample the event type topology according to the ratios of individual cross sections at the energy point M . hadrons 2.5.1 Sampling of M √ hadrons To calculate the total observed cross section at s, we split the integral of Eq. (1) into two parts, i.e. (cid:90) (cid:90) √ ≡ M0 2m σ (m) s 2m σ (m) σ(s) σI(s)+σII(s) = dm W (s,x)| −0 | + dm W (s,x)| −0 | , (9) s 1 Π(m) 2 s 1 Π(m) 2 Mth (cid:113) M0 where the threshold energy M is t√he sum of masses for the ﬁnal state particles, and the th − broken point is taken at M = s 2 sEcut with a cut Ecut on the ISR photon energy. At 0 γ γ ∼ the BESIII detector, the designed photon energy of the detection range is from 20 MeV to 4.2 GeV [46]. If the photon energy is less than 20 MeV, it will be impossible to reconstruct it in the detection simulation. Hence in practice, Ecut can be set to an energy less than the γ ∼ sensitivity of photon detection. e.g., Ecut = 1 MeV. In the range 0 Ecut, the ISR photon is γ γ 010201-6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: No.

text: X

text: Chinese

text: Physics

text: C

text: Vol.

text: xx,

text: No.

text: x

text: xxxxxx

text: by

text: ﬁtting

text: the

text: cross

text: section

text: mode

text: by

text: mode.

text: These

text: empirical

text: functions

text: are

text: used

text: in

text: the

text: generator

text: for

text: the

text: calculation

text: of

text: the

text: ISR

text: correction

text: factor

text: and

text: event

text: type

text: sampling.

text: The

text: angular

text: distribution

text: for

text: ISR

text: photons

text: is

text: implemented

text: according

text: to

text: Eq.

text: However,

text: angular

text: distributions

text: are

text: implemented

text: only

text: for

text: two-body

text: decays,

text: namely,

text: θ

text: for

text: PP

text: (where

text: P

text: is

text: a

text: pseudoscalar

text: meson)

text: modes,

text: and

text: θ

text: for

text: the

text: PV

text: (α

text: and

text: BB¯

text: modes,

text: where

text: V

text: is

text: a

text: vector

text: meson,

text: and

text: B

text: is

text: a

text: baryon.

text: The

text: angular

text: distribution

text: parameter

text: α

text: for

text: the

text: BB

text: mode

text: is

text: taken

text: as

text: the

text: quark

text: model

text: prediction

text: The

text: phase

text: space

text: model

text: is

text: used

text: for

text: multi-body

text: decays.

text: LUND

text: Area

text: Law

text: model

text: The

text: hadronic

text: events

text: produced

text: in

text: the

text: e+e

text: annihilation

text: are

text: evolved

text: as

text: follows.

text: As

text: the

text: ﬁrst

text: step,

text: a

text: quark-antiquark

text: (qq¯)

text: pair

text: is

text: produced

text: from

text: a

text: virtual

text: photon,

text: coupled

text: to

text: the

text: e+e

text: pair.

text: Then

text: the

text: qq¯

text: branching

text: proceeds

text: via

text: emitting

text: gluons,

text: and

text: further

text: develops

text: into

text: hadrons.

text: In

text: the

text: high

text: energy

text: region,

text: the

text: cluster

text: model

text: (e.g.

text: HERWIG

text: and

text: LUND

text: string

text: model

text: (e.g.

text: are

text: available

text: and

text: precise

text: enough

text: to

text: describe

text: the

text: hadronic

text: fragmentation

text: with

text: parameters

text: optimized

text: at

text: boson

text: Z

text: peak.

text: However,

text: in

text: the

text: intermediate

text: and

text: low

text: energy

text: region,

text: parameters

text: need

text: to

text: be

text: optimized

text: or

text: a

text: new

text: model

text: is

text: desirable

text: to

text: describe

text: the

text: light

text: quark

text: fragmentation.

text: In

text: the

text: tau-charm

text: energy

text: region,

text: the

text: LUARLW

text: model

text: has

text: been

text: proposed

text: to

text: estimate

text: the

text: multiplicity

text: distribution

text: for

text: primary

text: hadrons

text: produced

text: from

text: the

text: string

text: fragmentation.

text: The

text: probability

text: distribution

text: reads:

text: µn

text: P

text: exp[c

text: +c

text: (n

text: µ)+c

text: (n

text: n

text: n!

text: with

text: µ

text: α

text: β

text: exp(γ

text: s),

text: where

text: c

text: ,c

text: ,c

text: ,α,β

text: and

text: γ

text: are

text: parameters

text: to

text: be

text: tuned

text: with

text: data.

text: An

text: interface

text: to

text: access

text: the

text: LUARLW

text: model

text: is

text: designed

text: in

text: the

text: BesEvtGen

text: framework,

text: and

text: is

text: only

text: used

text: to

text: generate

text: the

text: primary

text: hadrons.

text: The

text: further

text: decays

text: into

text: light

text: hadrons

text: are

text: realized

text: with

text: BesEvtGen

text: Monte

text: Carlo

text: algorithm

text: The

text: event

text: sampling

text: proceeds

text: via

text: two

text: steps.

text: Firstly,

text: the

text: mass

text: of

text: the

text: hadron

text: system,

text: M

text: hadrons

text: is

text: sampled

text: according

text: to

text: the

text: distribution

text: of

text: the

text: observed

text: cross

text: section,

text: i.e.

text: dσ(s)/dm,

text: for

text: the

text: process

text: e+e

text: γ

text: X

text: according

text: to

text: Eq.

text: For

text: simplicity,

text: the

text: ISR

text: energy,

text: s

text: M

text: is

text: ISR

text: i

text: hadrons

text: imposed

text: on

text: a

text: single

text: photon.

text: The

text: second

text: step

text: is

text: to

text: sample

text: the

text: event

text: type

text: topology

text: according

text: to

text: the

text: ratios

text: of

text: individual

text: cross

text: sections

text: at

text: the

text: energy

text: point

text: M

text: hadrons

text: Sampling

text: of

text: M

text: hadrons

text: To

text: calculate

text: the

text: total

text: observed

text: cross

text: section

text: at

text: s,

text: we

text: split

text: the

text: integral

text: of

text: Eq.

text: into

text: two

text: parts,

text: i.e.

text: σ

text: (m)

text: s

text: σ

text: (m)

text: σ(s)

text: σI(s)+σII(s)

text: dm

text: W

text: (s,x)|

text: −0

text: dm

text: W

text: (s,x)|

text: −0

text: s

text: Π(m)

text: s

text: Π(m)

text: Mth

text: where

text: the

text: threshold

text: energy

text: M

text: is

text: t√he

text: sum

text: of

text: masses

text: for

text: the

text: ﬁnal

text: state

text: particles,

text: and

text: the

text: th

text: broken

text: point

text: is

text: taken

text: at

text: M

text: s

text: sEcut

text: with

text: a

text: cut

text: Ecut

text: on

text: the

text: ISR

text: photon

text: energy.

text: At

text: γ

text: γ

text: the

text: BESIII

text: detector,

text: the

text: designed

text: photon

text: energy

text: of

text: the

text: detection

text: range

text: is

text: from

text: MeV

text: to

text: GeV

text: If

text: the

text: photon

text: energy

text: is

text: less

text: than

text: MeV,

text: it

text: will

text: be

text: impossible

text: to

text: reconstruct

text: it

text: in

text: the

text: detection

text: simulation.

text: Hence

text: in

text: practice,

text: Ecut

text: can

text: be

text: set

text: to

text: an

text: energy

text: less

text: than

text: the

text: γ

text: sensitivity

text: of

text: photon

text: detection.

text: e.g.,

text: Ecut

text: MeV.

text: In

text: the

text: range

text: Ecut,

text: the

text: ISR

text: photon

text: is

text: γ

text: γ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: No. X Chinese Physics C Vol. xx, No. x (201x) xxxxxx 10 data to the MC distributions, which are generated with optimized parameters for all sets. We require that the parameters can produce MC distributions having the best ﬁt goodness quality χ2/N, where N is the total number of bins for calculating the χ2 values. The optimal values are given in Table 2. These values are only responsible for unknown processes other than the exclusive modes. For example, the parameter PARJ(15)=0 implies that exclusive modes have √ produce suﬃcient scalar mesons, so the LUARLW model forbids the scalar meson production. Figure 4 shows a comparison of data and MC distributions at s = 3.65 GeV, where the MC sample is produced with the optimized parameters. The agreement between them is satisfactory. To demonstrate the ﬂexibility of these parameters at low energy points, we generate MC at 3.06 GeV with the same parameters, and Fig. 5 shows comparisons between the data and MC simulation. The agreement between the data and MC distributions is acceptable. However, ¯ above the DD threshold, we check these parameters with the data taken at 4.26, 4.23 and 4.6 GeV, and we ﬁnd that the agreement between data and MC gets worse. This suggests that the ¯ optimized parameters are acceptable only below the DD threshold. To optimize parameters ¯ above the DD threshold, the charm meson decays will have to be added. √¯ To validate this set of parameters for the MC generation below the DD threshold, we compare the charged track multiplicity distributions at 14 energy points from s = 2.2324 to 3.671 GeV, as shown in Fig. 6. When extending this set of parameters from 3.65 GeV to low energy points, the agreement between the data and MC multiplicity distributions gets better. This is due to the fact that the total cross section equals the sum of the exclusive ones when approaching the energy 2.0 GeV, as shown in Fig. 2. Fig. 4. (color online) Comparison of data to MC distributions at 3.65 GeV, where the MC sample is produced with the optimized parameters: (a) multiplicity of charged tracks, (b) cosine of polar angle of charged tracks, (c) energy of charged tracks, (d) multiplicity of photon, (e) energy of photon, (f) cosine of polar angle of photons, (g) azimuthal distribution, (h) pseudorapidity and (i) thrust. Where the points with errors are data, and shaded histogram is MC distribution. 010201-10 ##LTLine## ##LTLine## ##LTLine##

text: No.

text: X

text: Chinese

text: Physics

text: C

text: Vol.

text: xx,

text: No.

text: x

text: xxxxxx

text: data

text: to

text: the

text: MC

text: distributions,

text: which

text: are

text: generated

text: with

text: optimized

text: parameters

text: for

text: all

text: sets.

text: We

text: require

text: that

text: the

text: parameters

text: can

text: produce

text: MC

text: distributions

text: having

text: the

text: best

text: ﬁt

text: goodness

text: quality

text: where

text: N

text: is

text: the

text: total

text: number

text: of

text: bins

text: for

text: calculating

text: the

text: χ2

text: values.

text: The

text: optimal

text: values

text: are

text: given

text: in

text: Table

text: These

text: values

text: are

text: only

text: responsible

text: for

text: unknown

text: processes

text: other

text: than

text: the

text: exclusive

text: modes.

text: For

text: example,

text: the

text: parameter

text: implies

text: that

text: exclusive

text: modes

text: have

text: produce

text: suﬃcient

text: scalar

text: mesons,

text: so

text: the

text: LUARLW

text: model

text: forbids

text: the

text: scalar

text: meson

text: production.

text: Figure

text: shows

text: a

text: comparison

text: of

text: data

text: and

text: MC

text: distributions

text: at

text: s

text: GeV,

text: where

text: the

text: MC

text: sample

text: is

text: produced

text: with

text: the

text: optimized

text: parameters.

text: The

text: agreement

text: between

text: them

text: is

text: satisfactory.

text: To

text: demonstrate

text: the

text: ﬂexibility

text: of

text: these

text: parameters

text: at

text: low

text: energy

text: points,

text: we

text: generate

text: MC

text: at

text: GeV

text: with

text: the

text: same

text: parameters,

text: and

text: Fig.

text: shows

text: comparisons

text: between

text: the

text: data

text: and

text: MC

text: simulation.

text: The

text: agreement

text: between

text: the

text: data

text: and

text: MC

text: distributions

text: is

text: acceptable.

text: However,

text: above

text: the

text: DD

text: threshold,

text: we

text: check

text: these

text: parameters

text: with

text: the

text: data

text: taken

text: at

text: and

text: GeV,

text: and

text: we

text: ﬁnd

text: that

text: the

text: agreement

text: between

text: data

text: and

text: MC

text: gets

text: worse.

text: This

text: suggests

text: that

text: the

text: optimized

text: parameters

text: are

text: acceptable

text: only

text: below

text: the

text: DD

text: threshold.

text: To

text: optimize

text: parameters

text: above

text: the

text: DD

text: threshold,

text: the

text: charm

text: meson

text: decays

text: will

text: have

text: to

text: be

text: added.

text: To

text: validate

text: this

text: set

text: of

text: parameters

text: for

text: the

text: MC

text: generation

text: below

text: the

text: DD

text: threshold,

text: we

text: compare

text: the

text: charged

text: track

text: multiplicity

text: distributions

text: at

text: energy

text: points

text: from

text: s

text: to

text: GeV,

text: as

text: shown

text: in

text: Fig.

text: When

text: extending

text: this

text: set

text: of

text: parameters

text: from

text: GeV

text: to

text: low

text: energy

text: points,

text: the

text: agreement

text: between

text: the

text: data

text: and

text: MC

text: multiplicity

text: distributions

text: gets

text: better.

text: This

text: is

text: due

text: to

text: the

text: fact

text: that

text: the

text: total

text: cross

text: section

text: equals

text: the

text: sum

text: of

text: the

text: exclusive

text: ones

text: when

text: approaching

text: the

text: energy

text: GeV,

text: as

text: shown

text: in

text: Fig.

text: Fig.

text: (color

text: online)

text: Comparison

text: of

text: data

text: to

text: MC

text: distributions

text: at

text: GeV,

text: where

text: the

text: MC

text: sample

text: is

text: produced

text: with

text: the

text: optimized

text: parameters:

text: (a)

text: multiplicity

text: of

text: charged

text: tracks,

text: (b)

text: cosine

text: of

text: polar

text: angle

text: of

text: charged

text: tracks,

text: (c)

text: energy

text: of

text: charged

text: tracks,

text: (d)

text: multiplicity

text: of

text: photon,

text: (e)

text: energy

text: of

text: photon,

text: (f)

text: cosine

text: of

text: polar

text: angle

text: of

text: photons,

text: (g)

text: azimuthal

text: distribution,

text: (h)

text: pseudorapidity

text: and

text: (i)

text: thrust.

text: Where

text: the

text: points

text: with

text: errors

text: are

text: data,

text: and

text: shaded

text: histogram

text: is

text: MC

text: distribution.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: the eﬀective monomer-monomer interaction. The free energy expression for the system is as follows: √ F 1 N 1 = α2 + + B + C, (9) k T α2 α3σ3 α6σ6 B √ where α = R /( Nσ) characterizes the extent of the swelling, N the degree of poly- g merization, σ the diameter of the monomeric unit, B and C are the second and the third virial coeﬃcients, respectively. The ﬁrst and the second terms in the above free energy are the entropic contributions with the the coiled and the globular states given by α > 1 and α < 1, respectively. The third and the fourth terms represent the energy contributions from the two-body and the three-body interactions, respectively. In general, the second virial coeﬃcient B is given by (cid:90) (cid:0) (cid:1) ∞ − B = 2π dr r2 1 − e Φ(r)/kBT , (10) 0 where Φ(r) is the monomer-monomer interaction potential. Here we assume that it has  the following form:   ∞ : r < σ Φ(r) = , (11) φ(r) : r > σ where Φ(r) for r < σ corresponds to the (hard-core) excluded volume interaction, and φ(r) for r > σ (see Eq. (6)) is the KW solvophobic potential. Substituting Eqs. (6) and (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (11) into Eq. (10), we obtain the second virial coeﬃcient B as 2πσ3 1 + Q 1 2 2 B = 1 + 3 − + , (12) 3 1 − Q ln L (ln L)2 (ln L)3 where L = (1 − S)/(1 + S). Hereafter we use the dimensionless quantities such as the temperature T˜ = k T/(u − w), B the distance r˜ = r/σ, the solvent mediated interaction potential φ˜ = φ/(u − w), and the second virial coeﬃcient B˜ = 3B/2πσ3. In Fig. 8, we plot B˜ as a function of T˜ for diﬀerent ˜ q-values. We observe that B remains almost unity for low temperatures and then decreases 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: eﬀective

text: monomer-monomer

text: interaction.

text: The

text: free

text: energy

text: expression

text: for

text: the

text: system

text: is

text: as

text: follows:

label: equation

text: F

text: N

text: α2

text: B

text: C,

text: k

text: T

text: α2

text: α3σ3

text: α6σ6

text: B

text: where

text: α

text: R

text: Nσ)

text: characterizes

text: the

text: extent

text: of

text: the

text: swelling,

text: N

text: the

text: degree

text: of

text: poly-

text: g

text: merization,

text: σ

text: the

text: diameter

text: of

text: the

text: monomeric

text: unit,

text: B

text: and

text: C

text: are

text: the

text: second

text: and

text: the

text: third

text: virial

text: coeﬃcients,

text: respectively.

text: The

text: ﬁrst

text: and

text: the

text: second

text: terms

text: in

text: the

text: above

text: free

text: energy

text: are

text: the

text: entropic

text: contributions

text: with

text: the

text: the

text: coiled

text: and

text: the

text: globular

text: states

text: given

text: by

text: α

text: and

text: α

text: respectively.

text: The

text: third

text: and

text: the

text: fourth

text: terms

text: represent

text: the

text: energy

text: contributions

text: from

text: the

text: two-body

text: and

text: the

text: three-body

text: interactions,

text: respectively.

text: In

text: general,

text: the

text: second

text: virial

text: coeﬃcient

text: B

text: is

text: given

text: by

text: B

text: 2π

text: dr

text: e

text: Φ(r)/kBT

text: where

text: Φ(r)

text: is

text: the

text: monomer-monomer

text: interaction

text: potential.

text: Here

text: we

text: assume

text: that

text: it

text: has

text: the

text: following

text: form:

text: r

text: σ

text: Φ(r)

text: φ(r)

text: r

text: σ

text: where

text: Φ(r)

text: for

text: r

text: σ

text: corresponds

text: to

text: the

text: (hard-core)

text: excluded

text: volume

text: interaction,

text: and

text: φ(r)

text: for

text: r

text: σ

text: (see

text: Eq.

text: is

text: the

text: KW

text: solvophobic

text: potential.

text: Substituting

text: Eqs.

text: and

text: into

text: Eq.

text: we

text: obtain

text: the

text: second

text: virial

text: coeﬃcient

text: B

text: as

text: 2πσ3

text: Q

text: B

text: Q

text: ln

text: L

text: (ln

text: (ln

text: where

text: L

text: S).

text: Hereafter

text: we

text: use

text: the

text: dimensionless

text: quantities

text: such

text: as

text: the

text: temperature

text: T˜

text: k

text: T/(u

text: w),

text: B

text: the

text: distance

text: r˜

text: r/σ,

text: the

text: solvent

text: mediated

text: interaction

text: potential

text: φ˜

text: φ/(u

text: w),

text: and

text: the

text: second

text: virial

text: coeﬃcient

text: B˜

text: In

text: Fig.

text: we

text: plot

text: B˜

text: as

text: a

text: function

text: of

text: T˜

text: for

text: diﬀerent

text: q-values.

text: We

text: observe

text: that

text: B

text: remains

text: almost

text: unity

text: for

text: low

text: temperatures

text: and

text: then

text: decreases

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: FIG. 2. Schematic representation for the theoretical model. Black and blue beads represent monomer and solvent molecules, respectively. (a) Single polymer chain in explicit solvent, (b) single polymer chain in implicit solvent where the eﬀect of solvent is incorporated in the monomer- monomer interaction potential φ(r), and (c) φ(r) modeled by the solvophobic potential given by KW.37 The potential assumes a one-dimensional solvent lattice where each solvent molecule has q states. Neighboring solvent molecules exist in a bounded state (BS) when both are in the state “1” and in unbounded state (US) state otherwise. Solute can occupy interstitial sites between bounded (BS) solvent molecules. spherically symmetric in nature and does not relate to a speciﬁc solvent or solute, which is in correspondence with the modeling framework employed in our simulations. The KW model belongs to a class of implicit solvent models40,41 which incorporate diﬀerent interaction energies depending on the proximity of the solvent to a solute molecule. These models have been used for studying the solubility of small solutes in water. The KW model is one of the simplest among these models as it has only two solvent interaction energies; one for the bound state and another for the bulk state. These interaction energies are analogous to the monomer-solvent interaction energy ((cid:15) ), and solvent-solvent interaction energy ((cid:15) ) AS SS in our simulation model, respectively. Hence it can be seen that the model contains those contributions which have been emphasized in our simulation results. In the KW-model, solvent molecules form a one-dimensional lattice with a nearest neigh- bor interaction, and each solvent molecule can exist in q diﬀerent states denoted by 1, 2, · · · q as shown in Fig. 2. The interaction energy between the neighboring solvent molecules is w when both of them are in the state “1”, and u otherwise with u > w. Here the former and the latter cases are termed as the bound state (BS) state and the unbound state (US) 6 ##LTFigure## ##LTFigure##

text: FIG.

text: Schematic

text: representation

text: for

text: the

text: theoretical

text: model.

text: Black

text: and

text: blue

text: beads

text: represent

text: monomer

text: and

text: solvent

text: molecules,

text: respectively.

text: (a)

text: Single

text: polymer

text: chain

text: in

text: explicit

text: solvent,

text: (b)

text: single

text: polymer

text: chain

text: in

text: implicit

text: solvent

text: where

text: the

text: eﬀect

text: of

text: solvent

text: is

text: incorporated

text: in

text: the

text: monomer-

text: monomer

text: interaction

text: potential

text: φ(r),

text: and

text: (c)

text: φ(r)

text: modeled

text: by

text: the

text: solvophobic

text: potential

text: given

text: by

text: The

text: potential

text: assumes

text: a

text: one-dimensional

text: solvent

text: lattice

text: where

text: each

text: solvent

text: molecule

text: has

text: q

text: states.

text: Neighboring

text: solvent

text: molecules

text: exist

text: in

text: a

text: bounded

text: state

text: (BS)

text: when

text: both

text: are

text: in

text: the

text: state

text: “1”

text: and

text: in

text: unbounded

text: state

text: (US)

text: state

text: otherwise.

text: Solute

text: can

text: occupy

text: interstitial

text: sites

text: between

text: bounded

text: (BS)

text: solvent

text: molecules.

text: spherically

text: symmetric

text: in

text: nature

text: and

text: does

text: not

text: relate

text: to

text: a

text: speciﬁc

text: solvent

text: or

text: solute,

text: which

text: is

text: in

text: correspondence

text: with

text: the

text: modeling

text: framework

text: employed

text: in

text: our

text: simulations.

text: The

text: KW

text: model

text: belongs

text: to

text: a

text: class

text: of

text: implicit

text: solvent

text: which

text: incorporate

text: diﬀerent

text: interaction

text: energies

text: depending

text: on

text: the

text: proximity

text: of

text: the

text: solvent

text: to

text: a

text: solute

text: molecule.

text: These

text: models

text: have

text: been

text: used

text: for

text: studying

text: the

text: solubility

text: of

text: small

text: solutes

text: in

text: water.

text: The

text: KW

text: model

text: is

text: one

text: of

text: the

text: simplest

text: among

text: these

text: models

text: as

text: it

text: has

text: only

text: two

text: solvent

text: interaction

text: energies;

text: one

text: for

text: the

text: bound

text: state

text: and

text: another

text: for

text: the

text: bulk

text: state.

text: These

text: interaction

text: energies

text: are

text: analogous

text: to

text: the

text: monomer-solvent

text: interaction

text: energy

text: and

text: solvent-solvent

text: interaction

text: energy

text: AS

text: SS

text: in

text: our

text: simulation

text: model,

text: respectively.

text: Hence

text: it

text: can

text: be

text: seen

text: that

text: the

text: model

text: contains

text: those

text: contributions

text: which

text: have

text: been

text: emphasized

text: in

text: our

text: simulation

text: results.

text: In

text: the

text: KW-model,

text: solvent

text: molecules

text: form

text: a

text: one-dimensional

text: lattice

text: with

text: a

text: nearest

text: neigh-

text: bor

text: interaction,

text: and

text: each

text: solvent

text: molecule

text: can

text: exist

text: in

text: q

text: diﬀerent

text: states

text: denoted

text: by

text: q

text: as

text: shown

text: in

text: Fig.

text: The

text: interaction

text: energy

text: between

text: the

text: neighboring

text: solvent

text: molecules

text: is

text: w

text: when

text: both

text: of

text: them

text: are

text: in

text: the

text: state

text: “1”,

text: and

text: u

text: otherwise

text: with

text: u

text: w.

text: Here

text: the

text: former

text: and

text: the

text: latter

text: cases

text: are

text: termed

text: as

text: the

text: bound

text: state

text: (BS)

text: state

text: and

text: the

text: unbound

text: state

text: (US)

text: ##LTFigure##

text: ##LTFigure##


text: FIG. 5. Representative snapshots of equilibrated N = 400 chain at (cid:15) = 1.7 for diﬀerent tem- AS peratures. Green and red beads represent solvophobic and amphiphilic units, respectively. (a) T˜ = 0.55, (b) T˜ = 0.6, (c) T˜ = 0.7 and (d) T˜ = 0.8. Solvent beads are not included for clarity. 550 ǫ = 1.4 ǫ = 1.7 400 (a) ǫAS = 1.8 ǫAS = 2.0 (b) ǫAS = 1.7 AS AS 350 500 300 450 250 200 400 0.50 0.55 0.60 0.65 0.70 0.5 0.6 0.7 0.8 T T FIG. 6. (a) Variation of N with T at diﬀerent values for (a) N = 200 chain at diﬀerent values of s (cid:15) , (b) N = 400 chain at (cid:15) =1.7. AS AS intermediate and high values, respectively. To further understand the behavior in these three states, we examine the variation of the number of bound solvent beads, N , with the s temperature. In Fig. 6(a), the number of bound solvent is found to be almost independent of T for low and high values of (cid:15) , whereas it decreases by increasing the temperature for AS intermediate values. In Fig. 6(b), we plot N for the N = 400 chain as a function of T , s when (cid:15) = 1.7. The decrease of N with the temperature is more prominent as compared AS s with that of the N = 200 chain case. Moreover, N markedly decreases around T = 0.65 s which coincides with the temperature around which R also decreases (see Fig. 4(b)). It g should be stressed that the change in R and N with the temperature is observed in our g s model with minimal interactions as in Eq. (1). It is important to note that even when the interaction parameters are independent of the temperature, the temperature dependence of R is induced by the bound solvent number N for a range of values of (cid:15) . g s AS 10 N s N s ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: Representative

text: snapshots

text: of

text: equilibrated

text: N

text: chain

text: at

text: for

text: diﬀerent

text: tem-

text: AS

text: peratures.

text: Green

text: and

text: red

text: beads

text: represent

text: solvophobic

text: and

text: amphiphilic

text: units,

text: respectively.

text: (a)

text: T˜

text: (b)

text: T˜

text: (c)

text: T˜

text: and

text: (d)

text: T˜

text: Solvent

text: beads

text: are

text: not

text: included

text: for

text: clarity.

text: ǫ

text: ǫ

text: (a)

text: ǫAS

text: ǫAS

text: (b)

text: ǫAS

text: AS

text: AS

text: T

text: T

text: FIG.

text: (a)

text: Variation

text: of

text: N

text: with

text: T

text: at

text: diﬀerent

text: values

text: for

text: (a)

text: N

text: chain

text: at

text: diﬀerent

text: values

text: of

text: s

text: (b)

text: N

text: chain

text: at

text: AS

text: AS

text: intermediate

text: and

text: high

text: values,

text: respectively.

text: To

text: further

text: understand

text: the

text: behavior

text: in

text: these

text: three

text: states,

text: we

text: examine

text: the

text: variation

text: of

text: the

text: number

text: of

text: bound

text: solvent

text: beads,

text: N

text: with

text: the

text: s

text: temperature.

text: In

text: Fig.

text: the

text: number

text: of

text: bound

text: solvent

text: is

text: found

text: to

text: be

text: almost

text: independent

text: of

text: T

text: for

text: low

text: and

text: high

text: values

text: of

text: whereas

text: it

text: decreases

text: by

text: increasing

text: the

text: temperature

text: for

text: AS

text: intermediate

text: values.

text: In

text: Fig.

text: we

text: plot

text: N

text: for

text: the

text: N

text: chain

text: as

text: a

text: function

text: of

text: T

text: s

text: when

text: The

text: decrease

text: of

text: N

text: with

text: the

text: temperature

text: is

text: more

text: prominent

text: as

text: compared

text: AS

text: s

text: with

text: that

text: of

text: the

text: N

text: chain

text: case.

text: Moreover,

text: N

text: markedly

text: decreases

text: around

text: T

text: s

text: which

text: coincides

text: with

text: the

text: temperature

text: around

text: which

text: R

text: also

text: decreases

text: (see

text: Fig.

text: It

text: g

text: should

text: be

text: stressed

text: that

text: the

text: change

text: in

text: R

text: and

text: N

text: with

text: the

text: temperature

text: is

text: observed

text: in

text: our

text: g

text: s

text: model

text: with

text: minimal

text: interactions

text: as

text: in

text: Eq.

text: It

text: is

text: important

text: to

text: note

text: that

text: even

text: when

text: the

text: interaction

text: parameters

text: are

text: independent

text: of

text: the

text: temperature,

text: the

text: temperature

text: dependence

text: of

text: R

text: is

text: induced

text: by

text: the

text: bound

text: solvent

text: number

text: N

text: for

text: a

text: range

text: of

text: values

text: of

text: g

text: s

text: AS

text: N

text: s

text: N

text: s

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

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text: A NEW GRAVITATIONAL-WAVE SIGNATURE OF SASI ACTIVITIES 3 Fig. 1.— In each set of panels, we plot, top; gravitational wave amplitude of plus mode A+ [cm], bottom; the characteristic wave strain in frequency-time domain h˜ in a logarithmic scale which is over plotted by the expected peak frequency F (black line denoted by “A”). peak “B” indicates the low frequency component. The component “A” is originated from the PNS g-mode oscillation (Marek & Janka 2009; Mu¨ller et al. 2013). The component “B” is considered to be associated with the SASI activities (see Sec. ). Left and right panels are for TM1 and SFHx, respectively. We mention that SFHx (left) and TM1 (right) are softer and stiﬀer EoS models, respectively. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: A

text: NEW

text: GRAVITATIONAL-WAVE

text: SIGNATURE

text: OF

text: SASI

text: ACTIVITIES

text: Fig.

text: In

text: each

text: set

text: of

text: panels,

text: we

text: plot,

text: top;

text: gravitational

text: wave

text: amplitude

text: of

text: plus

text: mode

text: A+

text: [cm],

text: bottom;

text: the

text: characteristic

text: wave

text: strain

text: in

text: frequency-time

text: domain

text: h˜

text: in

text: a

text: logarithmic

text: scale

text: which

text: is

text: over

text: plotted

text: by

text: the

text: expected

text: peak

text: frequency

text: F

text: (black

text: line

text: denoted

text: by

text: “A”).

text: peak

text: “B”

text: indicates

text: the

text: low

text: frequency

text: component.

text: The

text: component

text: “A”

text: is

text: originated

text: from

text: the

text: PNS

text: g-mode

text: oscillation

text: (Marek

text: Janka

text: Mu¨ller

text: et

text: al.

text: The

text: component

text: “B”

text: is

text: considered

text: to

text: be

text: associated

text: with

text: the

text: SASI

text: activities

text: (see

text: Sec.

text: Left

text: and

text: right

text: panels

text: are

text: for

text: and

text: SFHx,

text: respectively.

text: We

text: mention

text: that

text: SFHx

text: (left)

text: and

text: (right)

text: are

text: softer

text: and

text: stiﬀer

text: EoS

text: models,

text: respectively.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 12 Liping Xu (cid:104) (cid:90) (cid:90) (cid:105) which gives (cid:82) − E t 1 | − ∗ | lim K1 2/ν WK W (α) 2+2γ/νdαds = 0. →∞ s s K 0 0 Moreover, w(cid:104) (cid:90)e re(cid:90)call that a.s. 1 |WK − W∗(α)|γdα ≤(cid:105) 1 +(cid:90)C (cid:107)f (cid:107) as in Step 2, whence 0 s s γ,p s Lp E t 1 | − | | − ∗ | ≤ t E | − | (cid:107) (cid:107) WK X 2 WK W (α) γdαds [ WK X 2](1 + C f )ds. s s s s s s γ,p s Lp 0 0 0 → ∞ (cid:104) (cid:90) (cid:90) (cid:105) (cid:90) Letting K , by dominated convergence, we ﬁnd (recall (3.4)) E t 1 | − | | − ∗ | ≤ t (cid:107) (cid:107) lim sup WK X 2 WK W (α) γdαds J (1 + C f )ds. s s s s s γ,p s Lp (cid:2) (cid:0) K 0 0 0 (cid:1)(cid:3) (cid:2) (cid:0) ≥ (cid:1)(cid:3) ∈ ∈ (cid:55)→ Next, it is obvious that for each (cid:96) 1 ﬁxed, for all s [0, T], all α [0, 1], the function v | − ∗ | E | − min( v W (α) γ, (cid:96)) is bounded and continuous. By (3.3), we conclude that lim →∞ min WK W∗(α)|γ, (cid:96) s = E min |W (cid:104)−(cid:90) W(cid:90)∗(α)|γ, (cid:96) and, by domina(cid:0)ted convergence, th(cid:1)at, stK(cid:105)ill for (cid:96) ≥ 1 ﬁxeds, s s s lim(cid:90) E(cid:90) t 1 |W∗(α) − X∗((cid:2)α)|2 m(cid:0)in |WK − W∗(α)(cid:1)|(cid:3)γ, (cid:96) dαds →∞ s s s s K 0 0 = t 1 |W∗(α) − X∗(α)|2E min |W − W∗(α)|γ, (cid:96) (cid:82)dαds. s s s s 0 0 (cid:82) ∼ E | − ∗ | ≤ | ∗ − | ≤ BCuγ,tp(cid:107)sifnsc(cid:107)eLpWbsy(cid:104)((cid:90)2f.1s,)(cid:90).wFeuhrathvee,rmfoorreea, cwheαhaﬁvxeed0,1(cid:0)|W[ms∗i(nα()−WXs s∗(Wα)s|2(dα(cid:1)α) =γ, (cid:96)E)(cid:105)α] [|WsR∗(cid:90)3−WXss∗|(2α])= Wv 22γ(ffss(,dfv˜s)) ≤1J+s. All in all, we have checked that E t 1 | ∗ − ∗ | | − ∗ | ≤ t (cid:107) (cid:107) lim W (α) X (α) 2 min WK W (α) γ, (cid:96) dαds C J (1 + f )ds. →∞ s s s s s s Lp K 0 0 0 → ∞ (cid:90) ≥ Gathering all the previous estimates to let K in (3.7): for each (cid:96) 1 ﬁxed, ≤ W − t (cid:107) (cid:107) J 2(f , f˜ ) + C(T, f , f˜ , f)(cid:96) κ(γ) + C J (1 + f )ds. t 2 0 0 0 0 s s Lp 0 → ∞ (cid:18) (cid:90) (cid:0) (cid:1) (cid:19) Letting now (cid:96) and using the Gr¨onwall lemma, we ﬁnd ≤ W t (cid:107) (cid:107) J 2(f , f˜ ) exp C 1 + f ds . t 2 0 0 γ,p s Lp 0 W ≤ Since 2(f , f˜) J , this completes the proof. 2 t t t It remains to prove Proposition 3.1. We start with a technical result. ∈ − ∈ Lemma 3.3. Assume (1.3) for some γ ( 1, 0), some ν (0, 1) with γ + ν > 0 and recall that the deviation function c was deﬁned by (2.2). Consider f ∈ P (R3) and φ (x) = (2π(cid:15))−3/2e−|x|2/(2(cid:15)). Set ∗ 2 (cid:15) f(cid:15)(w) = (f φ )(w). (cid:15) (cid:90) (cid:90) (cid:90) (cid:90) ∈ R ∈ (cid:16) (cid:112) (cid:17) (i) There exists a constant C > 0 such that for all x 3, all (cid:15) (0, 1), ∞ − 2π | |φ (v x) ≤ | | c(v, v∗, z, ϕ) (cid:15) dϕdzf(dv)f(dv∗) C 1 + m (f) + x , R R f(cid:15)(x) 2 3 3 0 0 ∈ (ii) For al(cid:90)l ∈(cid:15) (cid:90) (0,(cid:90)1), a(cid:90)ll R > 0, there is(cid:12)a constant CR,(cid:15) > 0 (d(cid:12)epending only on m2(f)) such that for (cid:12) (cid:12) all x, y B(0, R), (cid:12) (cid:12) (cid:12) (cid:12) ∞ − − 2π | | φ (v x) − φ (v y) ≤ | − | c(v, v∗, z, ϕ) (cid:15) (cid:15) dϕdzf(dv)f(dv∗) C x y . R R f(cid:15)(x) f(cid:15)(y) R,(cid:15) 3 3 0 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Liping

text: Xu

text: which

text: gives

text: E

text: t

text: lim

text: WK

text: W

text: (α)

text: s

text: s

text: K

text: Moreover,

text: that

text: a.s.

text: |WK

text: W∗(α)|γdα

text: as

text: in

text: Step

text: whence

text: s

text: s

text: γ,p

text: s

text: Lp

text: E

text: t

text: t

text: E

text: WK

text: X

text: WK

text: W

text: (α)

text: γdαds

text: WK

text: X

text: C

text: f

text: )ds.

text: s

text: s

text: s

text: s

text: s

text: s

text: γ,p

text: s

text: Lp

text: Letting

text: K

text: by

text: dominated

text: convergence,

text: we

text: ﬁnd

text: (recall

text: E

text: t

text: t

text: lim

text: sup

text: WK

text: X

text: WK

text: W

text: (α)

text: γdαds

text: J

text: C

text: f

text: )ds.

text: s

text: s

text: s

text: s

text: s

text: γ,p

text: s

text: Lp

text: K

text: Next,

text: it

text: is

text: obvious

text: that

text: for

text: each

text: ﬁxed,

text: for

text: all

text: s

text: T],

text: all

text: α

text: the

text: function

text: v

text: E

text: min(

text: v

text: W

text: (α)

text: γ,

text: is

text: bounded

text: and

text: continuous.

text: By

text: we

text: conclude

text: that

text: lim

text: min

text: WK

text: W∗(α)|γ,

text: s

text: E

text: min

text: |W

text: and,

text: by

text: convergence,

text: for

text: ﬁxeds,

text: s

text: s

text: s

text: t

text: |W∗(α)

text: |WK

text: dαds

text: s

text: s

text: s

text: s

text: K

text: t

text: |W∗(α)

text: min

text: |W

text: W∗(α)|γ,

text: s

text: s

text: s

text: s

text: E

text: =γ,

text: Wv

text: All

text: in

text: all,

text: we

text: have

text: checked

text: that

text: E

text: t

text: t

text: lim

text: W

text: (α)

text: X

text: (α)

text: min

text: WK

text: W

text: (α)

text: γ,

text: dαds

text: C

text: J

text: f

text: )ds.

text: s

text: s

text: s

text: s

text: s

text: s

text: Lp

text: K

text: Gathering

text: all

text: the

text: previous

text: estimates

text: to

text: let

text: K

text: in

text: for

text: each

text: ﬁxed,

text: W

text: t

text: J

text: f˜

text: C(T,

text: f

text: f˜

text: κ(γ)

text: C

text: J

text: f

text: )ds.

text: t

text: s

text: s

text: Lp

text: Letting

text: now

text: and

text: using

text: the

text: Gr¨onwall

text: lemma,

text: we

text: ﬁnd

text: W

text: t

text: J

text: f˜

text: exp

text: C

text: f

text: ds

text: t

text: γ,p

text: s

text: Lp

text: W

text: Since

text: f˜)

text: J

text: this

text: completes

text: the

text: proof.

text: t

text: t

text: t

text: It

text: remains

text: to

text: prove

text: Proposition

text: We

text: start

text: with

text: a

text: technical

text: result.

text: Lemma

text: Assume

text: for

text: some

text: γ

text: some

text: ν

text: with

text: γ

text: ν

text: and

text: recall

text: that

text: the

text: deviation

text: function

text: c

text: was

text: deﬁned

text: by

text: Consider

text: f

text: P

text: and

text: φ

text: (x)

text: Set

text: (f

text: φ

text: )(w).

text: R

text: (i)

text: There

text: exists

text: a

text: constant

text: C

text: such

text: that

text: for

text: all

text: x

text: all

text: 2π

text: |φ

text: (v

text: x)

text: c(v,

text: v∗,

text: z,

text: ϕ)

text: dϕdzf(dv)f(dv∗)

text: C

text: m

text: (f)

text: x

text: R

text: R

text: (ii)

text: For

text: R

text: there

text: constant

text: only

text: on

text: such

text: that

text: for

text: all

text: x,

text: y

text: R),

text: 2π

text: φ

text: (v

text: x)

text: φ

text: (v

text: y)

text: c(v,

text: v∗,

text: z,

text: ϕ)

text: dϕdzf(dv)f(dv∗)

text: C

text: x

text: y

text: R

text: R

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Uniqueness and The Nanbu Particle System for Soft Potentials 17 L ∗ (cid:90) (cid:90) (cid:90) (cid:16) (cid:17) (cid:90) But since (X ) = f˜, this rewrites (recall (2.3)) α t t ∞ B 2π − A ψ(x) = ψ(x + c(x, v∗, z, ϕ) ψ(x) dϕdzf˜(dv∗) = ψ(x, v∗)f˜(dv∗). t R t R t 3 0 0 3 ≤ ≤ ≤ ≤ ≤ ≤ ∈ R We thus have to prove that for any 0 s ... s s t T, any ψ , ..., ψ C1( 3), and any ∈ R 1 k 1 k b ψ C1( 3), b E F [ (X)] = 0, where F : D([0, ∞), R3) (cid:55)→ R is d(cid:16)e(cid:89)ﬁned by (cid:17)(cid:16) (cid:90) (cid:17) F k − − t B (λ) = ψ (λ ) ψ(λ ) ψ(λ ) ψ(λ )dr . i si t s r r i=1 s We of course start from E[F(cid:15)n(X(cid:16)(cid:15)(cid:89)n)] = 0, wh(cid:17)e(cid:16)re, recalling (3.9),(cid:90) (cid:17) F k − − t A (λ) = ψ (λ ) ψ(λ ) ψ(λ ) ˜ ψ(λ )dr . (cid:15) i si t s r,(cid:15) r (cid:12) (cid:12) (cid:12) i=1 (cid:12) (cid:12) s (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) We then write E F ≤ E F − E F E F − E F [ (X)] [ (X)] [ (X(cid:15)n)] + [ (X(cid:15)n)] [ (X(cid:15)n)] . (cid:15)n (cid:55)→ A R ×R On the one hand, we know from [7, Lemma 3.3] that (x, v∗) ψ(x, v∗) is continuous on 3 3 and | − | F (cid:82) (cid:82) ∈ D ∞ R bounded by C x v∗ γ+1. We thus easily deduce that is continuous at each λ ([0, ), 3) which ∈ D ∞ R does not jump at s , ..., s , s, t (this is a.s. the case of X ([0, ), 3) because it has no deterministic 1 k |F | ≤ | − | tUimsinegjutmhapt b0y<thγe+A1ld<ous1,ctrhitaetriosunp).∈We aEls[osudpedu(cid:82)ce|Xth(cid:15)|a]t< ∞(λb)y StCep(14+an0dt rRe3caλllring vt∗haγt+1Xf˜r(cid:15)n(dgvo∗)edsri)n. |E(cid:15) F(0,1) − E[0F,T] t | → ∞ law(cid:12) to X, we easily conclude(cid:12) that [ (X)] [ (X(cid:15)n)] tends to 0 as n . (cid:12)(cid:12) On the other hand, since(cid:12)(cid:12)|F(λ) − F (λ)| ≤ C| t(B ψ(λ ) − A˜ ψ(λ ))dr| and X(cid:15) ∼ f˜(cid:15), (cid:15) s r r r,(cid:15) r r r E(cid:90)[F(X(cid:104)(cid:15)(cid:12)(cid:12)(cid:12)n(cid:90))] −(cid:90)E[F(cid:90)(cid:15)n(X(cid:90)(cid:15)n)] (cid:104) (cid:105) (cid:12)(cid:12)(cid:12)(cid:105) ∞ − ≤C (cid:90)st E(cid:12)(cid:12)(cid:12) (cid:90) (cid:90)R3 0(cid:90) 0(cid:90)2π (cid:90)R3 ψ(Xr(cid:15)n + c(v, v∗, z, ϕ)(cid:104)) φ(cid:15)fn˜r(cid:15)(nv(XrX(cid:15)nr(cid:15))n)f˜r(dv) − δXr(cid:15)n(dv) (cid:105)dϕdzf˜r(dv∗) dr(cid:12)(cid:12)(cid:12) ∞ =C t (cid:82)2π (cid:82) ψ(w + c(v, v∗, z, ϕ)) φ (v − w)f˜ (dv)(cid:82)− f˜(cid:15)n(w)δ (dv) dwdϕdzf˜ (dv∗) dr. (cid:82) (cid:82) s R3 0 0 R3 R3 (cid:15)n r r w r But we ca(cid:12)(cid:12)n write R3 R3 ψ(w +−c(v(cid:12)(cid:12), v∗, z, ϕ))f˜r(cid:15)n(w)δw(dv)dw = R3 ψ(w + c(w, v∗, z, ϕ))f˜r(cid:15)n(w)dw = R3 R3 ψ(w(cid:12) + c(w, v∗, z, ϕ))φ(cid:15)n(v w(cid:12))f˜r(dv)dw, so that E(cid:90)[F((cid:12)(cid:12)(cid:12)X(cid:90)(cid:15)n)(cid:90)] − E(cid:90)[F(cid:15)n((cid:90)X(cid:15)n(cid:90))] (cid:104) (cid:105) ∞ ≤ t 2π − (cid:12) C ψ(w + c(v, v∗, z, ϕ)) ψ(w + c(w, v∗, z, ϕ)) (cid:12)(cid:12) R R R (cid:90)s (cid:12) (cid:90) 3 (cid:90)0 (cid:90)0 (cid:90) 3 (cid:90) 3 (cid:104) − (cid:12) φ (v w)f˜ (dv)dwdϕdzf˜ (dv∗) dr (cid:12) (cid:15)n r r ∞ t 2π (cid:105) − − (cid:12) =C R R R ψ(w + c(v, v∗, z, ϕ + ϕ0(v v∗, w v∗))) (cid:12)(cid:12) s 3 0 0 3 3 − − ψ(w + c(w, v∗, z, ϕ)) φ (v w)f˜ (dv)dwdϕdzf˜ (dv∗) dr. (cid:90) (cid:104) (cid:15)n r r(cid:105) The last equality uses the 2π-periodicity of c. We now put − − − − R (v, v∗, z, ϕ) := ψ(w + c(v, v∗, z, ϕ + ϕ (v v∗, w v∗))) ψ(w + c(w, v∗, z, ϕ)) φ (v w)dw, n R 0 (cid:15)n 3 and show the following two things: ##LTLine##

text: Uniqueness

text: and

text: The

text: Nanbu

text: Particle

text: System

text: for

text: Soft

text: Potentials

text: L

text: But

text: since

text: (X

text: f˜,

text: this

text: rewrites

text: (recall

text: α

text: t

text: t

text: B

text: 2π

text: A

text: ψ(x)

text: ψ(x

text: c(x,

text: v∗,

text: z,

text: ϕ)

text: ψ(x)

text: dϕdzf˜(dv∗)

text: ψ(x,

text: v∗)f˜(dv∗).

text: t

text: R

text: t

text: R

text: t

text: R

text: We

text: thus

text: have

text: to

text: prove

text: that

text: for

text: any

text: s

text: s

text: s

text: t

text: T,

text: any

text: ψ

text: ψ

text: and

text: any

text: R

text: k

text: k

text: b

text: ψ

text: b

text: E

text: F

text: (X)]

text: where

text: F

text: R

text: is

text: by

text: F

text: k

text: t

text: B

text: (λ)

text: ψ

text: (λ

text: ψ(λ

text: ψ(λ

text: ψ(λ

text: )dr

text: i

text: si

text: t

text: s

text: r

text: r

text: s

text: We

text: of

text: course

text: start

text: from

text: recalling

text: F

text: k

text: t

text: A

text: (λ)

text: ψ

text: (λ

text: ψ(λ

text: ψ(λ

text: ψ(λ

text: )dr

text: i

text: si

text: t

text: s

text: r

text: s

text: We

text: then

text: write

text: E

text: F

text: E

text: F

text: E

text: F

text: E

text: F

text: E

text: F

text: (X)]

text: (X)]

text: A

text: R

text: ×R

text: On

text: the

text: one

text: hand,

text: we

text: know

text: from

text: Lemma

text: that

text: (x,

text: v∗)

text: ψ(x,

text: v∗)

text: is

text: continuous

text: on

text: and

text: F

text: D

text: R

text: bounded

text: by

text: C

text: x

text: v∗

text: We

text: thus

text: easily

text: deduce

text: that

text: is

text: continuous

text: at

text: each

text: λ

text: which

text: D

text: R

text: does

text: not

text: jump

text: at

text: s

text: s

text: s,

text: t

text: (this

text: is

text: a.s.

text: the

text: case

text: of

text: X

text: because

text: it

text: has

text: no

text: deterministic

text: k

text: |F

text: tUimsinegjutmhapt

text: ∞(λb)y

text: t

text: to

text: X,

text: we

text: easily

text: that

text: (X)]

text: tends

text: to

text: as

text: n

text: On

text: the

text: other

text: hand,

text: F

text: (λ)|

text: C|

text: t(B

text: ψ(λ

text: A˜

text: ψ(λ

text: ))dr|

text: and

text: s

text: r

text: r

text: r

text: r

text: r

text: ≤C

text: c(v,

text: v∗,

text: z,

text: =C

text: t

text: ψ(w

text: c(v,

text: v∗,

text: z,

text: ϕ))

text: φ

text: (v

text: w)f˜

text: (dv)

text: dwdϕdzf˜

text: (dv∗)

text: dr.

text: s

text: r

text: r

text: w

text: r

text: But

text: we

text: write

text: ψ(w

text: v∗,

text: z,

text: ψ(w

text: c(w,

text: v∗,

text: z,

text: c(w,

text: v∗,

text: z,

text: so

text: that

text: t

text: 2π

text: C

text: ψ(w

text: c(v,

text: v∗,

text: z,

text: ϕ))

text: ψ(w

text: c(w,

text: v∗,

text: z,

text: ϕ))

text: R

text: R

text: R

text: φ

text: (v

text: w)f˜

text: (dv)dwdϕdzf˜

text: (dv∗)

text: dr

text: r

text: r

text: t

text: 2π

text: =C

text: R

text: R

text: R

text: ψ(w

text: c(v,

text: v∗,

text: z,

text: ϕ

text: v∗,

text: w

text: v∗)))

text: s

text: ψ(w

text: c(w,

text: v∗,

text: z,

text: ϕ))

text: φ

text: (v

text: w)f˜

text: (dv)dwdϕdzf˜

text: (dv∗)

text: dr.

text: r

text: The

text: last

text: equality

text: uses

text: the

text: of

text: c.

text: We

text: now

text: put

text: R

text: (v,

text: v∗,

text: z,

text: ϕ)

text: ψ(w

text: c(v,

text: v∗,

text: z,

text: ϕ

text: ϕ

text: (v

text: v∗,

text: w

text: v∗)))

text: ψ(w

text: c(w,

text: v∗,

text: z,

text: ϕ))

text: φ

text: (v

text: w)dw,

text: n

text: R

text: and

text: show

text: the

text: following

text: two

text: things:

text: ##LTLine##