text: 5 (cid:112) (cid:38) it is suﬃcient to take√Φ 3.3 m . In Eq. (2.15) H = 4π/3 (M/m )Φ is the initial value of the Hubble in Pl in Pl in parameter and t = 2 3π Φ /(Mm ) is the time moment when Φ = 0, see (2.14). To be more precise, inﬂation end in Pl ∼ ends when the Hubble parameter becomes appro(cid:112)ximately equal to the mass of the inﬂaton, Hfin M, because at this moment the exponential expansion terminated and the inﬂaton ﬁeld started to oscillate around zero with ∼ frequency M. Correspondingly the exponential expansion turned into the matter dominated one, a t2/3. So the ﬁnal amplitude of Φ should be taken as Φ = 3/(4π) m . Correspondingly the ﬁnal time would be slightly fin Pl changed. In order to avoid too large density perturbations one should choose the value of the inﬂaton mass M in the − (cid:46) (cid:46) − − range 10 7 M/m 10 6; accordingly we take M = 10 6 m . Pl Pl To arrange the desired scenario we should set the value of Φ still at inﬂationary stage such that the distances 0 between diﬀerent domains and the domain sizes exponentially expanded up to cosmologically large scales. We can make a naive estimate of the necessary duration of inﬂation after the inﬂaton ﬁeld reached value Φ = Φ . 0 Suppose that at the present time the size of a domain is about 10 Mpc. If the characteristic scale of the initial ∼ energy/temperature is of the order of T, then it is natural to expect the domain size to be of the order of l 1/T. ∼ Due to regular cosmological expansion it would increase to z = (1/T)(T/3K) 0.1 cm. So during inﬂation the ∼ · domains should be expanded by a factor of (10 Mpc)/(0.1 cm) 3 1026. Therefore, we require that after Φ passes · (cid:39) Φ the inﬂation should last at least 60 e-foldings, N = ln (3 1026) 60, 0 (cid:90) e tend 2πΦ2 (cid:38) (cid:39) H(t)dt = 0 (cid:114) N 60, (2.16) m2 e Pl t0 Φ N ≈ 0 = e 3.1, (2.17) m 2π Pl where t is the time when Φ = Φ . However, since the process initiated at inﬂationary stage, when temperature 0 0 in classical sense did not exist, the characteristic scale of the seed of the domain should be of the order of the inverse ∼ ∼ ≈ Hubble parameter, l 1/H 1/M. So the obtained above result would be shifted to N 60 + ln(M/T ), ≈ − ÷ − heat where T is the universe temperature after inﬂation. Since typically T (10 5 10 6) m the estimate heat heat Pl presented above remains practically unchanged. Depending upon the value of Φ, there are three diﬀerent regimes of the evolution of the ﬁeld χ: − (cid:29) 1. Initial stage: Φ > Φ and (Φ Φ ) Φ . 0 (cid:112) 0 1 The mass of χ is supposed to be small in comparison with the expansion rate, m < H, so ﬁeld χ sits near the minimum of its potential at χ = 0 (see left plot in Fig. 2), with the dispersion of the order of (cid:104) (cid:105) ∼ { } χ2 min H4/m2, H2/ λ (see e.g. [20]). χ | − | (cid:46) 2. Second stage: Φ Φ Φ . 0 1 − ≈ − At this stage the squared eﬀective mass of χ becomes negative: m2 2µ2V (Φ) 2µ2V (Φ) < 0, if m is − small enough. The minima of the potential (2.10) move from zero as η(t) or η(t), where (cid:0) (cid:1) 1 − η2(t) = 2µ2V (Φ) m2 . (2.18) λ χ To have equal amplitudes of both positive and negative CP violation we need to impose the condition ± that χ reaches one or other minimum χ = η(t) and stays there (approximately). This minimum moves (cid:112) (cid:29) − exponentially fast and χ would follow it e.g. if µ H, since Eq. (2.7) becomes then χ¨ 2µ2χV (Φ) = 0, ∝ − (cid:29) therefore χ(t) grows roughly as exponent, χ exp(µt). For M = 10 6m , Φ = 3.1 m we get µ H = ∼ · − Pl 0 Pl 4π/3 M/m Φ 6 10 6m . Pl Pl But even when χ grows exponentially there still should be enough time for χ to reach the minimum η(t). If − the inﬂaton Φ goes from Φ + 2Φ to Φ 2Φ during the interval τ, we should require that 0 1 0 1 √ 8 3πΦ (cid:38) η 1 2µ2 µτ = µ 1 ln max = ln , (2.19) Mm χ 2 λ χ2 Pl in χ in (cid:38) M√m 2µ2 µ Pl ln , (2.20) 16 3πΦ1 λχχ2in ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: it

text: is

text: suﬃcient

text: to

text: take√Φ

text: m

text: In

text: Eq.

text: H

text: (M/m

text: )Φ

text: is

text: the

text: initial

text: value

text: of

text: the

text: Hubble

text: in

text: Pl

text: in

text: Pl

text: in

text: parameter

text: and

text: t

text: 3π

text: Φ

text: /(Mm

text: is

text: the

text: time

text: moment

text: when

text: Φ

text: see

text: To

text: be

text: more

text: precise,

text: inﬂation

text: end

text: in

text: Pl

text: ends

text: when

text: the

text: Hubble

text: parameter

text: becomes

text: equal

text: to

text: the

text: mass

text: of

text: the

text: inﬂaton,

text: Hfin

text: M,

text: because

text: at

text: this

text: moment

text: the

text: exponential

text: expansion

text: terminated

text: and

text: the

text: inﬂaton

text: ﬁeld

text: started

text: to

text: oscillate

text: around

text: zero

text: with

text: frequency

text: M.

text: Correspondingly

text: the

text: exponential

text: expansion

text: turned

text: into

text: the

text: matter

text: dominated

text: one,

text: a

text: So

text: the

text: ﬁnal

text: amplitude

text: of

text: Φ

text: should

text: be

text: taken

text: as

text: Φ

text: m

text: Correspondingly

text: the

text: ﬁnal

text: time

text: would

text: be

text: slightly

text: fin

text: Pl

text: changed.

text: In

text: order

text: to

text: avoid

text: too

text: large

text: density

text: perturbations

text: one

text: should

text: choose

text: the

text: value

text: of

text: the

text: inﬂaton

text: mass

text: M

text: in

text: the

text: range

text: M/m

text: accordingly

text: we

text: take

text: M

text: m

text: Pl

text: Pl

text: To

text: arrange

text: the

text: desired

text: scenario

text: we

text: should

text: set

text: the

text: value

text: of

text: Φ

text: still

text: at

text: inﬂationary

text: stage

text: such

text: that

text: the

text: distances

text: between

text: diﬀerent

text: domains

text: and

text: the

text: domain

text: sizes

text: exponentially

text: expanded

text: up

text: to

text: cosmologically

text: large

text: scales.

text: We

text: can

text: make

text: a

text: naive

text: estimate

text: of

text: the

text: necessary

text: duration

text: of

text: inﬂation

text: after

text: the

text: inﬂaton

text: ﬁeld

text: reached

text: value

text: Φ

text: Φ

text: Suppose

text: that

text: at

text: the

text: present

text: time

text: the

text: size

text: of

text: a

text: domain

text: is

text: about

text: Mpc.

text: If

text: the

text: characteristic

text: scale

text: of

text: the

text: initial

text: energy/temperature

text: is

text: of

text: the

text: order

text: of

text: T,

text: then

text: it

text: is

text: natural

text: to

text: expect

text: the

text: domain

text: size

text: to

text: be

text: of

text: the

text: order

text: of

text: l

text: Due

text: to

text: regular

text: cosmological

text: expansion

text: it

text: would

text: increase

text: to

text: z

text: cm.

text: So

text: during

text: inﬂation

text: the

text: domains

text: should

text: be

text: expanded

text: by

text: a

text: factor

text: of

text: cm)

text: Therefore,

text: we

text: require

text: that

text: after

text: Φ

text: passes

text: Φ

text: the

text: inﬂation

text: should

text: last

text: at

text: least

text: e-foldings,

text: N

text: ln

text: e

text: tend

text: 2πΦ2

text: H(t)dt

text: N

text: e

text: Pl

text: Φ

text: N

text: e

text: m

text: 2π

text: Pl

text: where

text: t

text: is

text: the

text: time

text: when

text: Φ

text: Φ

text: However,

text: since

text: the

text: process

text: initiated

text: at

text: inﬂationary

text: stage,

text: when

text: temperature

text: in

text: classical

text: sense

text: did

text: not

text: exist,

text: the

text: characteristic

text: scale

text: of

text: the

text: seed

text: of

text: the

text: domain

text: should

text: be

text: of

text: the

text: order

text: of

text: the

text: inverse

text: Hubble

text: parameter,

text: l

text: So

text: the

text: obtained

text: above

text: result

text: would

text: be

text: shifted

text: to

text: N

text: ln(M/T

text: heat

text: where

text: T

text: is

text: the

text: universe

text: temperature

text: after

text: inﬂation.

text: Since

text: typically

text: T

text: m

text: the

text: estimate

text: heat

text: heat

text: Pl

text: presented

text: above

text: remains

text: practically

text: unchanged.

text: Depending

text: upon

text: the

text: value

text: of

text: Φ,

text: there

text: are

text: three

text: diﬀerent

text: regimes

text: of

text: the

text: evolution

text: of

text: the

text: ﬁeld

text: χ:

text: Initial

text: stage:

text: Φ

text: Φ

text: and

text: (Φ

text: Φ

text: Φ

text: The

text: mass

text: of

text: χ

text: is

text: supposed

text: to

text: be

text: small

text: in

text: comparison

text: with

text: the

text: expansion

text: rate,

text: m

text: H,

text: so

text: ﬁeld

text: χ

text: sits

text: near

text: the

text: minimum

text: of

text: its

text: potential

text: at

text: χ

text: (see

text: left

text: plot

text: in

text: Fig.

text: with

text: the

text: dispersion

text: of

text: the

text: order

text: of

text: χ2

text: min

text: λ

text: (see

text: e.g.

text: χ

text: Second

text: stage:

text: Φ

text: Φ

text: Φ

text: At

text: this

text: stage

text: the

text: squared

text: eﬀective

text: mass

text: of

text: χ

text: becomes

text: negative:

text: (Φ)

text: (Φ)

text: if

text: m

text: is

text: small

text: enough.

text: The

text: minima

text: of

text: the

text: potential

text: move

text: from

text: zero

text: as

text: η(t)

text: or

text: η(t),

text: where

text: (Φ)

text: λ

text: χ

text: To

text: have

text: equal

text: amplitudes

text: of

text: both

text: positive

text: and

text: negative

text: CP

text: violation

text: we

text: need

text: to

text: impose

text: the

text: condition

text: that

text: χ

text: reaches

text: one

text: or

text: other

text: minimum

text: χ

text: η(t)

text: and

text: stays

text: there

text: (approximately).

text: This

text: minimum

text: moves

text: exponentially

text: fast

text: and

text: χ

text: would

text: follow

text: it

text: e.g.

text: if

text: µ

text: H,

text: since

text: Eq.

text: becomes

text: then

text: χ¨

text: (Φ)

text: therefore

text: χ(t)

text: grows

text: roughly

text: as

text: exponent,

text: χ

text: exp(µt).

text: For

text: M

text: Φ

text: m

text: we

text: get

text: µ

text: H

text: Pl

text: Pl

text: M/m

text: Φ

text: Pl

text: Pl

text: But

text: even

text: when

text: χ

text: grows

text: exponentially

text: there

text: still

text: should

text: be

text: enough

text: time

text: for

text: χ

text: to

text: reach

text: the

text: minimum

text: η(t).

text: If

text: the

text: inﬂaton

text: Φ

text: goes

text: from

text: Φ

text: 2Φ

text: to

text: Φ

text: 2Φ

text: during

text: the

text: interval

text: τ,

text: we

text: should

text: require

text: that

text: 3πΦ

text: η

text: µτ

text: µ

text: ln

text: max

text: ln

text: Mm

text: χ

text: λ

text: χ2

text: Pl

text: in

text: χ

text: in

text: M√m

text: µ

text: Pl

text: ln

text: 3πΦ1

text: λχχ2in

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (Dated: May 27, 2020) The Boltzmann equation framework for inelastic Maxwell models is considered to determine the transport coeﬃcients associated with the mass, momentum and heat ﬂuxes of a granular binary mixture in spatially inhomogeneous states close to the simple shear ﬂow. The Boltzmann equation is solved by means of a Chapman-Enskog-like expansion around the (local) shear ﬂow distributions f(0) for each species that retain all the hydrodynamic orders in the shear rate. Due to the anisotropy r induced by the shear ﬂow, tensorial quantities are required to describe the transport processes instead of the conventional scalar coeﬃcients. These tensors are given in terms of the solutions of a set of coupled equations, which can be analytically solved as functions of the shear rate a, the coeﬃcients of restitution α and the parameters of the mixture (masses, diameters and composition). Since rs the reference distribution functions f(0) apply for arbitrary values of the shear rate and are not r restricted to weak dissipation, the corresponding generalized coeﬃcients turn out to be nonlinear functions of both a and α . The dependence of the relevant elements of the three diﬀusion tensors rs on both the shear rate and dissipation is illustrated in the tracer limit case, the results showing that the deviation of the generalized transport coeﬃcients from their forms for vanishing shear rates is in general signiﬁcant. A comparison with the previous results obtained analytically for inelastic hard spheres by using Grad’s moment method is carried out showing a good agreement over a wide range of values for the coeﬃcients of restitution. Finally, as an application of the theoretical expressions derived here for the transport coeﬃcients, thermal diﬀusion segregation of an intruder immersed in a granular gas is also studied. PACS numbers: 05.20.Dd, 45.70.Mg, 51.10.+y I. INTRODUCTION transport coeﬃcients. Due to the mathematical diﬃcul- ties met in obtaining those coeﬃcients from the Boltz- mann collision operator for IHS [1], the inelastic version Granular media under rapid ﬂow conditions are of the BGK model [6] was considered to determine the amenable to a fruitful modelization through a gas of in- above generalized transport coeﬃcients [7, 8]. On the elastic hard spheres (IHS) [1]. In the simplest model, other hand, explicit expressions for these coeﬃcients were the grains are assumed to be smooth so that the inelas- derived by assuming particular perturbations where the ticity is characterized through a constant (positive) co- ≤ steady state conditions of the USF apply [7, 8]. This al- eﬃcient of normal restitution α 1 that only aﬀects lowed us to perform a linear stability analysis of the hy- the translational degrees of freedom of the grains. The drodynamic equations with respect to the USF state [8] case α = 1 corresponds to elastic collisions. Due to the to get the conditions for instability at long wavelengths. kinetic-energy dissipation in collisions, energy must be The results derived for IHS from the BGK model has externally injected to the granular gas in order to main- been then revisited by considering a mean ﬁeld version tain it in rapid ﬂow regime (ﬂuid-like description). In of the hard sphere system where randomly chosen pairs some cases, the system is driven into the ﬂow through a of particles collide with a random impact direction. This (linear) shear ﬁeld (simple or uniform shear ﬂow, USF) assumption, which yields a Boltzmann collision operator where a steady state is achieved when the energy dissi- with a collision rate independent of the relative veloc- pated by collisions is balanced by the energy supplied by ity of the two colliding particles, opens the possibility shearing work. The study of the rheological properties of obtaining exact results for granular gases in the con- in the steady USF has received consequential attention text of the Boltzmann kinetic equation. The above in- in the past years [2, 3], especially in the case of monodis- teraction model is referred to as the inelastic Maxwell perse granular gases. model (IMM) [9–16] and it has been widely considered The USF state is deﬁned by a constant density n, a by physicists and mathematicians alike in the past few uniform granular temperature T, and a linear velocity years to unveil in a clean way the role of dissipation in proﬁle u = ay, where a is the constant shear rate. In x granular ﬂows. In particular, the use of IMM allows us the steady state, the system admits a non-Newtonian to analytically determine the set of generalized transport description [4, 5] characterized by shear-rate dependent coeﬃcients around the USF state for general unsteady viscosity and normal stress diﬀerences. An interesting conditions [17]. problem is the analysis of momentum and heat trans- port in spatially inhomogeneous states close to the USF. All the above results refer to monocomponent granu- The physical situation is such that the granular gas is lar gases. However, a real granular system is generally in a strongly sheared state that deviates from the USF characterized by some degree of polydispersity in den- conditions by small spatial gradients. The response of sity and size (granular mixtures). Needless to say, the the system to these perturbations gives rise to additional diﬃculties for obtaining explicit expressions of the trans- contributions to the momentum and heat ﬂuxes, which port coeﬃcients increase considerably when one consid- can be characterized by generalized shear-rate dependent ers multicomponent systems since not only the number

text: (Dated:

label: paragraph

text: May

label: date

text: The

label: abstract

text: Boltzmann

text: equation

text: framework

text: for

text: inelastic

text: Maxwell

text: models

text: is

text: considered

text: to

text: determine

text: the

text: transport

text: coeﬃcients

text: associated

text: with

text: the

text: mass,

text: momentum

text: and

text: heat

text: ﬂuxes

text: of

text: a

text: granular

text: binary

text: mixture

text: in

text: spatially

text: inhomogeneous

text: states

text: close

text: to

text: the

text: simple

text: shear

text: ﬂow.

text: The

text: Boltzmann

text: equation

text: is

text: solved

text: by

text: means

text: of

text: a

text: Chapman-Enskog-like

text: expansion

text: around

text: the

text: (local)

text: shear

text: ﬂow

text: distributions

text: for

text: each

text: species

text: that

text: retain

text: all

text: the

text: hydrodynamic

text: orders

text: in

text: the

text: shear

text: rate.

text: Due

text: to

text: the

text: anisotropy

text: r

text: induced

text: by

text: the

text: shear

text: ﬂow,

text: tensorial

text: quantities

text: are

text: required

text: to

text: describe

text: the

text: transport

text: processes

text: instead

text: of

text: the

text: conventional

text: scalar

text: coeﬃcients.

text: These

text: tensors

text: are

text: given

text: in

text: terms

text: of

text: the

text: solutions

text: of

text: a

text: set

text: of

text: coupled

text: equations,

text: which

text: can

text: be

text: analytically

text: solved

text: as

text: functions

text: of

text: the

text: shear

text: rate

text: a,

text: the

text: coeﬃcients

text: of

text: restitution

text: α

text: and

text: the

text: parameters

text: of

text: the

text: mixture

text: (masses,

text: diameters

text: and

text: composition).

text: Since

text: rs

text: the

text: reference

text: distribution

text: functions

text: apply

text: for

text: arbitrary

text: values

text: of

text: the

text: shear

text: rate

text: and

text: are

text: not

text: r

text: restricted

text: to

text: weak

text: dissipation,

text: the

text: corresponding

text: generalized

text: coeﬃcients

text: turn

text: out

text: to

text: be

text: nonlinear

text: functions

text: of

text: both

text: a

text: and

text: α

text: The

text: dependence

text: of

text: the

text: relevant

text: elements

text: of

text: the

text: three

text: diﬀusion

text: tensors

text: rs

text: on

text: both

text: the

text: shear

text: rate

text: and

text: dissipation

text: is

text: illustrated

text: in

text: the

text: tracer

text: limit

text: case,

text: the

text: results

text: showing

text: that

text: the

text: deviation

text: of

text: the

text: generalized

text: transport

text: coeﬃcients

text: from

text: their

text: forms

text: for

text: vanishing

text: shear

text: rates

text: is

text: in

text: general

text: signiﬁcant.

text: A

text: comparison

text: with

text: the

text: previous

text: results

text: obtained

text: analytically

text: for

text: inelastic

text: hard

text: spheres

text: by

text: using

text: Grad’s

text: moment

text: method

text: is

text: carried

text: out

text: showing

text: a

text: good

text: agreement

text: over

text: a

text: wide

text: range

text: of

text: values

text: for

text: the

text: coeﬃcients

text: of

text: restitution.

text: Finally,

text: as

text: an

text: application

text: of

text: the

text: theoretical

text: expressions

text: derived

text: here

text: for

text: the

text: transport

text: coeﬃcients,

text: thermal

text: diﬀusion

text: segregation

text: of

text: an

text: intruder

text: immersed

text: in

text: a

text: granular

text: gas

text: is

text: also

text: studied.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: transport

text: coeﬃcients.

text: Due

text: to

text: the

text: mathematical

text: diﬃcul-

text: ties

text: met

text: in

text: obtaining

text: those

text: coeﬃcients

text: from

text: the

text: Boltz-

text: mann

text: collision

text: operator

text: for

text: IHS

text: the

text: inelastic

text: version

text: Granular

text: media

text: under

text: rapid

text: ﬂow

text: conditions

text: are

text: of

text: the

text: BGK

text: model

text: was

text: considered

text: to

text: determine

text: the

text: amenable

text: to

text: a

text: fruitful

text: modelization

text: through

text: a

text: gas

text: of

text: in-

text: above

text: generalized

text: transport

text: coeﬃcients

text: On

text: the

text: elastic

text: hard

text: spheres

text: (IHS)

text: In

text: the

text: simplest

text: model,

text: other

text: hand,

text: explicit

text: expressions

text: for

text: these

text: coeﬃcients

text: were

text: the

text: grains

text: are

text: assumed

text: to

text: be

text: smooth

text: so

text: that

text: the

text: inelas-

text: derived

text: by

text: assuming

text: particular

text: perturbations

text: where

text: the

text: ticity

text: is

text: characterized

text: through

text: a

text: constant

text: (positive)

text: co-

text: steady

text: state

text: conditions

text: of

text: the

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text: 12 ratio γ on b∗oth the mole fraction x1 and the (reduced) 1.5 (a) shear rate a . Thus, for the sake of simplicity, we con- → sider the tracer limit (x 0) where the mass ﬂux is the 1.0 1 D */D * relevant ﬂux since the momentum and heat ﬂuxes of the xx 0 0.5 system (intruder plus gas particles) are the same as those previously obtained [17] for a monocomponent granular 0.0 gas of IMM. D */D * It must be remarked that a non-equilibrium phase -0.5 xy 0 transition has been recently [42] identiﬁed in the tracer limit for a granular binary mixture of IMM. This tran- -1.0 sition refers to the existence of a region (coined as the 0.0 0.2 0.4 0.6 0.8 1.0 ordered phase) where the contribution of tracer parti- cles to the total kinetic energy of the system is ﬁnite. a * 1.0 However, the above (surprising) behavior has been only (b) analytically found when the collision frequency ω is as- rs sumed to be independent of the temperature ratio (“plain vanilla Maxwell model”) and hence, it does not seem to 0.5 D */D * exist for the more realistic version of the IMM considered yy 0 here. The eﬀects of the above transition on the Navier- Stokes transport coeﬃcients of a granular binary mixture 0.0 has been recently studied [43]. D */D * A. Model A yx 0 (cid:39) In the tracer limit, P(0) P(0) and the relevant ele- -0.5 2 0.0 0.2 0.4 0.6 0.8 1.0 ments of the partial pressure tensor P(0) admit simpli- 1 ﬁed forms (see Appendix ). In particular, in the tracer a* (cid:39) ∗ ∗ limit, γ γ, ∂ P(0) = ∂ γ = ∂ ζ = ∂ ∗ζ = 0 and 1 x1 ij x1 x1 a FIG. 1: (Color online) Panel (a): Shear-rate dependence ∂ P(0) = P(0) /x . Taking into account these simpliﬁ- ∗ ∗ ∗ ∗ x1 1,ij 1,ij 1 of the (dimensionless) coeﬃcients Dxx/D0 and Dxy/D0 for ca(cid:18)tions, and for m(cid:19)odel A (β = 0), Eqs. (60)–(61) become d = 3, σ /σ = 1, m /m = 2 and three diﬀerent values of 1 2 1 2 the (common) coeﬃcient of restitution α: α = 1 (solid line), ∗ α = 0.8 (dashed red line) and α = 0.6 (dash-dotted blue line). 2a ∗ ∗ ∗ − ∗ ∗ − ∗ ∗ − − ∗ P + ζ D ν D a D = x 1P , Panel (b):∗Shea∗r-rate d∗epend∗ence of the (dimensionless) coef- d 2,xy ij D ij ik kj 1 1,ij ﬁcients D /D and D /D for the same parameter values. yy 0 yx 0 (78) These results pertain to model A (β = 0). (cid:20) (cid:18) (cid:19) (cid:21) ∗ ∗ ∗ hence, they vanish in the tracer limit) while D is in- 2a ∗ − ∗ ∂P ∗ ∗ − ∗ ∗ ij ∗ d P2,xy a ∂a2,∗xy + 2ζ Dp,ij νDDp,ij dependent∗ of x1. In addition, the derivatives ∂a∗P2,ij and ∂ ∗P appearing in Eqs. (78)–(80) are obtained in (cid:18) (cid:19) ∗ a 1,ij − ∗ ∗ ∗ − ∗ − ∗ ∂P Appendix . aikDp,kj = x1µP2,ij P1,ij x1µ a ∂a2∗,ij As in the case of IHS [29], the coeﬃcients D decouple ∗ ∗ ∗ ij +a∗ ∂∂Pa1∗,ij + 2ad 2 ∂P∂a2,∗xy − ζ∗ DT∗,ij, (79) fsrtoramighthtfeorowtahredrlyo.nTesheainrdex(cid:32)hpernecssei,onthseayrecan be obtained(cid:33) (cid:18) (cid:19) − ∗ ∗ ∗ x 1 ∗ − a P D = ∗ − ∗ 1 ∗ − ∗ P ∗ − ik∗ 1∗,kj − ∗ . ∗ ∗ ij ν 2a P ζ 1,ij ν 2a P ζ 2a ∗ ∗ ∂P ∗ (cid:18)− ∗ ∗ − ∗ (cid:19)∗ D d 2,xy D d 2,xy P + a 2,∗xy D ν D a D (81) d 2,xy ∂a T,ij D T,ij ik T,kj ∗ ∗ ∗ ∗ ∗ ∗ ∗ In the steady state ((2a /d)P2,xy + ζ = 0), Eq. (81) = x1µ a∗ ∂∂Pa2∗,ij − a∗ ∂∂Pa1∗,ij − 2ad 2 ∂P∂a2,∗xy − ζ∗ Dp∗,ij. idsercosnhseiasrteﬂnotww[i2th0].prTehveioruesmraeisnuilntgs dcoeeriﬃvecdienfotsr DIM∗M aunnd- ∗ p,ij D are coupled and they obey the set of simple al- (80) T,ij gebraic equations (79)–(80). As alluded to above, they are proportional to the concentration x and thus van- ∗ − ∗ ∗ 1 Here, ζ = (1 α2 )/2d and ν = (ω µ (1+α ))/d ish in the tracer limit. Yet, it is of interest to nor- ∗ ≡ 22 D 12 21 12 where ω ω /ν . Upon deriving Eq. (78) we have ne- malize them by their vanishing shear rate counterparts 12 12 0 ∗ glected the contributions coming from the tensors D (which are also proportional to x ), to study their depen- ∗ p,ij 1 and D since both tensors are proportional to x (and dence on parameters other than x . In order to illustrate T,ij 1 1 ##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## 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text: 23 ory of a Simple Monatomic Gas (New York, Academic New York, 1984). S. Chapman and T. G. Cowling, The Press, 1980). V. Garz´o and A. Santos, Kinetic Theory of Mathematical Theory of Nonuniform Gases (Cambridge Gases in Shear Flows. Nonlinear Transport (Dordrecht, University Press, Cambridge, 1970). F. Vega Reyes, V. Kluwer–Academic, 2003). R. Yano, J. Phys. A: Math. Garzo´ and A. Santos, Phys. Rev. E 75, 061306 (2007). V. Theor. 46, 375502 (2013). V. Garzo´, Phys. Rev. E 66, Garzo´ and E. Trizac, Europhys. Lett. 94, 50009 (2011); 021308 (2002). V. Garzo´, J. Stat. Mech. P02012 (2007). Phys. Rev. E 85, 011302 (2012). V. Garzo´, N. Khalil and H. Grad, Commun. Pure Appl. Math. 2, 331 (1949). V. E. Trizac, Eur. Phys. J. E 38, 16 (2015). See for instance Garzo´ and F. Vega Reyes, J. Stat. Mech. P07024 (2010). the special issue Focus on granular segregation, published See for instance, J. M. Montanero and V. Garzo´, Gran. in New J. Phys. (2013); editorial by K. E. Daniels and Matt. 4, 17 (2002); A. Barrat and E. Trizac, Gran. Matt. M. Schro¨ter, New. J . Phys. 15, 035017 (2013). See for 4, 57 (2002); Phys. Rev. E 66, 051303 (2002); S. R. Dahl, instance, S.Sarkar and D. V. Khakhar, Europhys. Lett. C. M. Hrenya, V. Garz´o, and J. W. Dufty, Phys. Rev. E 83, 54004 (2008); L. A. Golick and K. E. Daniels, Phys. 66, 041301 (2002); R. Pagnani, U. M. B. Marconi and Rev. E 80, 042301 (2009); L. B. H. May, L.A. Golick, K. A. Puglisi, Phys. Rev. E 66, 051304 (2002); R. Clelland C. Phillips, M. Shearer, and K. E. Daniels, Phys. Rev. E and C. M. Hrenya, Phys. Rev. E 65, 031301 (2002); J. 81, 051301 (2010); L. B. H. May, M. Shearer and K. E. M. Montanero and V. Garz´o, Phys. Rev. E 67, 021308 Daniels, J. Nonlinear Sci. 20, 689 (2010); K. M. Hill and (2003); P. E. Krouskop and J. Talbot, Phys. Rev. E 68, D. S. Tan, J. Fluid Mech. 756, 54 (2014). B. Arnarson 021304 (2003); H. Q. Wang, G. J. Jin and Y. Q. Ma, and J. T. Jenkins, Phys. Fluids 16, 4543 (2004). J. Kin- Phys. Rev. E 68, 031301 (2003); J. J. Brey, M. J. Ruiz- caid, E. G. D. Cohen, and M. Lo´pez de Haro, J. Chem. Montero and F. Moreno, Phys. Rev. Lett. 95, 098001 Phys. 86, 963 (1987). F. Vega Reyes, V. Garzo´ and N. (2005); Phys. Rev. E 73, 031301 (2006); M. Schro¨ter, S. Khalil, Phys. Rev. E 89, 052206 (2014). V. Garzo´, Phys. Ulrich, J. Kreft, J. B. Swift and H. L. Swinney, Phys. Rev. E 78, 020301(R) (2008); Eur. Phys. J. E 29, 261 Rev. E 74, 011307 (2006). R. D. Wildman and D. J. (2009); V. Garzo´ and F. Vega Reyes, Phys. Rev. E 85, Parker, Phys. Rev. Lett. 88, 064301 (2002); K. Feitosa 021308 (2012). F. Vega Reyes, A. Santos and V. Garzo´, and N. Menon, Phys. Rev. Lett. 88, 198301 (2002). V. Phys. Rev. Lett. 104, 028001 (2010). Thermal diﬀu- Garzo´ and J. W. Dufty, Phys. Rev. E 60, 5706 (1999). V. sion is induced by a thermal gradient; in the USF state, Garzo´ and A. Santos, J. Phys. A: Math. Theor. 40, 14927 temperature is uniform and one cannot determine the (2007). A. W. Lees and S. F. Edwards, J. Phys. C 5, 1921 thermal diﬀusion factor. We thus consider a state where (1972). J. M. Montanero and V. Garz´o, Physica A 310, there is a coexistence between a (linear) shear ﬁeld and 17 (2002). J. W. Dufty, A. Santos, J. J. Brey and R. F. a temperature gradient. With the resulting uniform heat Rodr´ıguez, Phys. Rev. A 33, 459 (1986). S. R. de Groot ﬂux, the steady condition (90) still applies. and P. Mazur, Nonequilibrium Thermodynamics (Dover,

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text: 8 (cid:18) (cid:19) A − 2 A − (cid:32)∂ L A M(cid:33)A (cid:32) (cid:33) aP(0) + ζ(0) (p∂ + T∂ ) ac 1 + + = A dp xy p T 1 y ∂c 1 1 1 2 1 x 2a ∂P(0) ∂ζ(0) B 2aT ∂P(0) ∂ζ(0) C xy xy + + p + + T , (39) d ∂x ∂x 1 dp ∂x ∂x 1 1 1 1 1 (cid:18) (cid:19) (cid:32) (cid:33) B − 2 B − 2a ∂P(cid:34)(0) (cid:32) ∂ζ(0) B(cid:33)− ∂ (cid:35) L B M B aP(0) + ζ(0) (p∂ + T∂ ) xy + ζ(0) + p ac 1 + + dp xy p T 1 d ∂p ∂p 1 y ∂c 1 1 1 2 x − 2aT − ∂P(0) − T ∂ζ(0) C = B P(0) p xy , (40) 1 dp2 xy ∂p p ∂p 1 (cid:18) (cid:19) (cid:32) (cid:33) C − 2 C − 2a 2aT ∂P(cid:32)(0) ∂ζ(0) C −(cid:33) ∂ aP(0) + ζ(0) (p∂ + T∂ ) P(0) + ζ(0) + xy + T ac 1 dp xy p T 1 dp xy dp ∂T ∂T 1 y ∂c x L C M C 2a ∂P(0) ∂ζ(0) B xy + + = C + + p , (41) 1 1 1 2 1 d ∂T ∂T 1 (cid:18) (cid:19) D − 2 D − ∂ − D L D M D aP(0) + ζ(0) (p∂ + T∂ ) ac 1,k(cid:96) aδ + + = D . (42) dp xy p T 1,k(cid:96) y ∂c ky 1,x(cid:96) 1 1,k(cid:96) 1 2,k(cid:96) 1,k(cid:96) x Here, A , B , C and D are deﬁned by Eqs. (A3)– 1 1 1 1,k(cid:96) L M (A6), respectively. Moreover, and are the lin- 1 1 earized Boltzmann collision operators around the refer- (cid:16) (cid:17) − ∂δu ence USF state: P(1) = η (cid:96) , (45) ij ijk(cid:96) ∂r k L − X = J [f(0), X] + J [X, f(0)] + J [X, f(0)] , 1 11 1 11 1 12 2 (43a) − (cid:48)(cid:48) ∂x − ∂p − ∂T q(1) = T2D 1 L λ , (46) i ij ∂r ij ∂r ij ∂r j j j M − X = J [f(0), X]. (43b) 1 12 2 where (cid:90) ↔ A similar equation for f(1) applies by setting 1 2. It is important to note tha2t for β = 1, Eqs. (39)–(42) D = − ρ dc c A (c), (47) are expected to have the same structu2re as that of the ij nm2 i 1,j Boltzmann equation for IHS, except for the explicit form (cid:90) L M of the operators and . s s Once the form of the distributions f(1) is known, the r −pm B ﬁrst-order corrections to the mass ﬂux j1(1,i), the pressure Dp,ij = ρ 1 dc ci 1,j(c), (48) tensor P(1) and the heat ﬂux q(1) can be obtained. They ij i (cid:90) are given by j1(1,i) = −m1mρ 2nDij ∂∂xr1 − ρpDp,ij ∂∂rp − Tρ DT,ij ∂∂rT , DT,ij = −Tmρ 1 dc ci C1,j(c), (49) j j j (44) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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