text: 2 was a representative of the modiﬁed level set method with The reason of making a choice is the inclusion of ther- excellent numerical performance and an adaptive mesh modynamics into the scheme34. It is theoretically shown reﬁnement extension of GFM can be found in Ref. 18. that the non-oscillatory property in pressure can be GFM can also be used to deﬁne interface conditions in preserved across material interfaces. Although this pa- the front-tracking method19. per takes two-ﬂuid ﬂows to illustrate the method, the There are many other multi-ﬂuid approaches avail- proposed algorithms can be applied to multi-ﬂuid ﬂow able in literature, such as the volume-of-ﬂuid (VOF) models even when computed over unstructured meshes. method20–22, the moment-of-ﬂuid (MOF) method23 and To demonstrate the performance of the proposed the BGK-based model24. The VOF method solves the schemes, we carry out several numerical experiments evolution equations for volume fractions, the mass and typically for illustrating the simulation of compressible energy equations for individual ﬂuids, and the momen- ﬂuid ﬂows. They are a two-ﬂuid compression problem in tum equation for the ﬂuid mixture, and it was improved order to show the necessity of kinetic energy exchange in Ref. 22 to eliminate pressure oscillations by deﬁning during ﬂuid mixture; the shock-interface interaction, and mixtures inside each cell with diﬀerent temperatures and the shock-bubble interaction problems. solving an extra one-phase energy equation. Ton’s ap- We organize this paper as follows. In Section II, we proach relieved the need to solve the evolution equations describe basic models for immiscible compressible multi- for volume fractions. The MOF method, another nu- ﬂuid ﬂows and discuss the cause of pressure errors. We merical approach, was proposed25 by using the volume provide the motivation of our non-oscillatory scheme in fraction and centroid for a more accurate representa- Section III, and propose the numerical method in Section tion of the material conﬁguration, interfaces and volume IV. To display the performance of the current method, we advection. provide several typical numerical results in the context of multi-ﬂuid ﬂows including the interaction of shock- Lagrangian and ALE frameworks are preferable in the interface and the interaction of shock-bubble in Section compressible multi-ﬂuid ﬂows due to their sharp captur- V. ing ability of material interfaces. There are a lot of contri- II. PHYSICAL MODELS FOR TWO-FLUID FLOWS butions on Lagrangian and ALE schemes, see Refs. 26–28 and references therein. Still, an indispensable ingredient has to be added in order to avoid the occurrence of pres- Under the assumption that all ﬂuid variables are de- sure oscillations at material interfaces, such as a pressure scribed by a single density ρ, a single pressure p and relaxation technique in the Lagrangian Godunov method a common ﬂuid velocity u, the Euler equations repre- with Tipton’s closure model26. A comparative study of senting conservation of mass, momentum and energy for multi-ﬂuid Lagrangian and Eulerian methods was made inviscid compressible multi-ﬂuid ﬂows take the form in Ref. 29, and a relaxation-projection method by La- grangian plus remapping the ﬂow quantities to the Eule- ∂ ∇ · rian mesh was designed in Ref. 30. ρ + (ρu) = 0, (1) ∂t It is worth noting that any non-conservative scheme may converge to wrong solutions31, providing incorrect internal energy or shock wave positions. Hence, re- ∂ ∇ · ⊗ searchers hope to simulate compressible multi-ﬂuid ﬂows (ρu) + (ρu u + pI) = 0, (2) using conservative schemes in the Eulerian framework. ∂t For this purpose, we propose an algorithm in this paper, (cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:18) (cid:19) (cid:21) based on the Godunov method, that can prevent non- physical oscillations at interfaces without sacriﬁcing the ∂ 1| | ∇ · 1| | numerical results of the compressible ﬂow phenomena ρ e + u 2 + ρu e + u 2 + pu = 0, ∂t 2 2 involving shock waves and rarefaction waves. Inspired (3) by Ref. 22, we solve an equation of mass fraction and a one-phase energy equation coupling with the Euler where e is the speciﬁc internal energy speciﬁed by an equations. In order to simulate kinetic energy more ac- EOS for the mixture. For two immiscible ﬂuids a and curately, we add a one-phase momentum equation for b, we denote z , φ , ρ , p , e , T , C , γ as their volume correction. As far as the interaction of shocks and in- k k k k k k v,k k fraction, mass fraction, density, pressure, speciﬁc internal terfaces is concerned, the exchange of kinetic energy is energy, temperature, speciﬁc heat capacity at constant processed. Motivated by Ref. 16, we use the isobaric volume and ratio of speciﬁc heats, respectively with k = equilibrium condition to compute the volume fractions a, b. Principles of volume average tell us of the mixture and the ratios of speciﬁc heats inside mixed ﬂuid cells. rule of thermodynamical parameters A beneﬁt of this method is its conservative form, which allows to suit for the ﬁnite volume framework. The re- sulting scheme is of Godunov-type, and an second order ρ = z ρ + z ρ , ρe = z ρ e + z ρ e . (4) a a b b a a a b b b accurate extension is made by using the space-time cou- pled generalized Riemann problem (GRP) solver32,33. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: was

text: a

text: representative

text: of

text: the

text: modiﬁed

text: level

text: set

text: method

text: with

text: The

text: reason

text: of

text: making

text: a

text: choice

text: is

text: the

text: inclusion

text: of

text: ther-

text: excellent

text: numerical

text: performance

text: and

text: an

text: adaptive

text: mesh

text: modynamics

text: into

text: the

text: It

text: is

text: theoretically

text: shown

text: reﬁnement

text: extension

text: of

text: GFM

text: can

text: be

text: found

text: in

text: Ref.

text: that

text: the

text: non-oscillatory

text: property

text: in

text: pressure

text: can

text: be

text: GFM

text: can

text: also

text: be

text: used

text: to

text: deﬁne

text: interface

text: conditions

text: in

text: preserved

text: across

text: material

text: interfaces.

text: Although

text: this

text: pa-

text: the

text: front-tracking

text: per

text: takes

text: two-ﬂuid

text: ﬂows

text: to

text: illustrate

text: the

text: method,

text: the

text: There

text: are

text: many

text: other

text: multi-ﬂuid

text: approaches

text: avail-

text: proposed

text: algorithms

text: can

text: be

text: applied

text: to

text: multi-ﬂuid

text: ﬂow

text: able

text: in

text: literature,

text: such

text: as

text: the

text: volume-of-ﬂuid

text: (VOF)

text: models

text: even

text: when

text: computed

text: over

text: unstructured

text: meshes.

text: the

text: moment-of-ﬂuid

text: (MOF)

text: and

text: To

text: demonstrate

text: the

text: performance

text: of

text: the

text: proposed

text: the

text: BGK-based

text: The

text: VOF

text: method

text: solves

text: the

text: schemes,

text: we

text: carry

text: out

text: several

text: numerical

text: experiments

text: evolution

text: equations

text: for

text: volume

text: fractions,

text: the

text: mass

text: and

text: typically

text: for

text: illustrating

text: the

text: simulation

text: of

text: compressible

text: energy

text: equations

text: for

text: individual

text: ﬂuids,

text: and

text: the

text: momen-

text: ﬂuid

text: ﬂows.

text: They

text: are

text: a

text: two-ﬂuid

text: compression

text: problem

text: in

text: tum

text: equation

text: for

text: the

text: ﬂuid

text: mixture,

text: and

text: it

text: was

text: improved

text: order

text: to

text: show

text: the

text: necessity

text: of

text: kinetic

text: energy

text: exchange

text: in

text: Ref.

text: to

text: eliminate

text: pressure

text: oscillations

text: by

text: deﬁning

text: during

text: ﬂuid

text: mixture;

text: the

text: shock-interface

text: interaction,

text: and

text: mixtures

text: inside

text: each

text: cell

text: with

text: diﬀerent

text: temperatures

text: and

text: the

text: shock-bubble

text: interaction

text: problems.

text: solving

text: an

text: extra

text: one-phase

text: energy

text: equation.

text: Ton’s

text: ap-

text: We

text: organize

text: this

text: paper

text: as

text: follows.

text: In

text: Section

text: II,

text: we

text: proach

text: relieved

text: the

text: need

text: to

text: solve

text: the

text: evolution

text: equations

text: describe

text: basic

text: models

text: for

text: immiscible

text: compressible

text: multi-

text: for

text: volume

text: fractions.

text: The

text: MOF

text: method,

text: another

text: nu-

text: ﬂuid

text: ﬂows

text: and

text: discuss

text: the

text: cause

text: of

text: pressure

text: errors.

text: We

text: merical

text: approach,

text: was

text: by

text: using

text: the

text: volume

text: provide

text: the

text: motivation

text: of

text: our

text: non-oscillatory

text: scheme

text: in

text: fraction

text: and

text: centroid

text: for

text: a

text: more

text: accurate

text: representa-

text: Section

text: III,

text: and

text: propose

text: the

text: numerical

text: method

text: in

text: Section

text: tion

text: of

text: the

text: material

text: conﬁguration,

text: interfaces

text: and

text: volume

text: IV.

text: To

text: display

text: the

text: performance

text: of

text: the

text: current

text: method,

text: we

text: advection.

text: provide

text: several

text: typical

text: numerical

text: results

text: in

text: the

text: context

text: of

text: multi-ﬂuid

text: ﬂows

text: including

text: the

text: interaction

text: of

text: shock-

text: Lagrangian

text: and

text: ALE

text: frameworks

text: are

text: preferable

text: in

text: the

text: interface

text: and

text: the

text: interaction

text: of

text: shock-bubble

text: in

text: Section

text: compressible

text: multi-ﬂuid

text: ﬂows

text: due

text: to

text: their

text: sharp

text: captur-

text: V.

text: ing

text: ability

text: of

text: material

text: interfaces.

text: There

text: are

text: a

text: lot

text: of

text: contri-

text: II.

text: PHYSICAL

text: MODELS

text: FOR

text: TWO-FLUID

text: FLOWS

text: butions

text: on

text: Lagrangian

text: and

text: ALE

text: schemes,

text: see

text: Refs.

text: and

text: references

text: therein.

text: Still,

text: an

text: indispensable

text: ingredient

text: has

text: to

text: be

text: added

text: in

text: order

text: to

text: avoid

text: the

text: occurrence

text: of

text: pres-

text: Under

text: the

text: assumption

text: that

text: all

text: ﬂuid

text: variables

text: are

text: de-

text: sure

text: oscillations

text: at

text: material

text: interfaces,

text: such

text: as

text: a

text: pressure

text: scribed

text: by

text: a

text: single

text: density

text: ρ,

text: a

text: single

text: pressure

text: p

text: and

text: relaxation

text: technique

text: in

text: the

text: Lagrangian

text: Godunov

text: method

text: a

text: common

text: ﬂuid

text: velocity

text: u,

text: the

text: Euler

text: equations

text: repre-

text: with

text: Tipton’s

text: closure

text: A

text: comparative

text: study

text: of

text: senting

text: conservation

text: of

text: mass,

text: momentum

text: and

text: energy

text: for

text: multi-ﬂuid

text: Lagrangian

text: and

text: Eulerian

text: methods

text: was

text: made

text: inviscid

text: compressible

text: multi-ﬂuid

text: ﬂows

text: take

text: the

text: form

text: in

text: Ref.

text: and

text: a

text: relaxation-projection

text: method

text: by

text: La-

text: grangian

text: plus

text: remapping

text: the

text: ﬂow

text: quantities

text: to

text: the

text: Eule-

text: rian

text: mesh

text: was

text: designed

text: in

text: Ref.

text: ρ

text: (ρu)

text: ∂t

text: It

text: is

text: worth

text: noting

text: that

text: any

text: non-conservative

text: scheme

text: may

text: converge

text: to

text: wrong

text: providing

text: incorrect

text: internal

text: energy

text: or

text: shock

text: wave

text: positions.

text: Hence,

text: re-

text: searchers

text: hope

text: to

text: simulate

text: compressible

text: multi-ﬂuid

text: ﬂows

text: (ρu)

text: (ρu

text: u

text: pI)

text: using

text: conservative

text: schemes

text: in

text: the

text: Eulerian

text: framework.

text: ∂t

text: For

text: this

text: purpose,

text: we

text: propose

text: an

text: algorithm

text: in

text: this

text: paper,

text: based

text: on

text: the

text: Godunov

text: method,

text: that

text: can

text: prevent

text: non-

text: physical

text: oscillations

text: at

text: interfaces

text: without

text: sacriﬁcing

text: the

text: numerical

text: results

text: of

text: the

text: compressible

text: ﬂow

text: phenomena

text: ρ

text: e

text: u

text: ρu

text: e

text: u

text: pu

text: ∂t

text: involving

text: shock

text: waves

text: and

text: rarefaction

text: waves.

text: Inspired

text: by

text: Ref.

text: we

text: solve

text: an

text: equation

text: of

text: mass

text: fraction

text: and

text: a

text: one-phase

text: energy

text: equation

text: coupling

text: with

text: the

text: Euler

text: where

text: e

text: is

text: the

text: speciﬁc

text: internal

text: energy

text: speciﬁed

text: by

text: an

text: equations.

text: In

text: order

text: to

text: simulate

text: kinetic

text: energy

text: more

text: ac-

text: EOS

text: for

text: the

text: mixture.

text: For

text: two

text: immiscible

text: ﬂuids

text: a

text: and

text: curately,

text: we

text: add

text: a

text: one-phase

text: momentum

text: equation

text: for

text: b,

text: we

text: denote

text: z

text: φ

text: ρ

text: p

text: e

text: T

text: C

text: γ

text: as

text: their

text: volume

text: correction.

text: As

text: far

text: as

text: the

text: interaction

text: of

text: shocks

text: and

text: in-

text: k

text: k

text: k

text: k

text: k

text: k

text: v,k

text: k

text: fraction,

text: mass

text: fraction,

text: density,

text: pressure,

text: speciﬁc

text: internal

text: terfaces

text: is

text: concerned,

text: the

text: exchange

text: of

text: kinetic

text: energy

text: is

text: energy,

text: temperature,

text: speciﬁc

text: heat

text: capacity

text: at

text: constant

text: processed.

text: Motivated

text: by

text: Ref.

text: we

text: use

text: the

text: isobaric

text: volume

text: and

text: ratio

text: of

text: speciﬁc

text: heats,

text: respectively

text: with

text: k

text: equilibrium

text: condition

text: to

text: compute

text: the

text: volume

text: fractions

text: a,

text: b.

text: Principles

text: of

text: volume

text: average

text: tell

text: us

text: of

text: the

text: mixture

text: and

text: the

text: ratios

text: of

text: speciﬁc

text: heats

text: inside

text: mixed

text: ﬂuid

text: cells.

text: rule

text: of

text: thermodynamical

text: parameters

text: A

text: beneﬁt

text: of

text: this

text: method

text: is

text: its

text: conservative

text: form,

text: which

text: allows

text: to

text: suit

text: for

text: the

text: ﬁnite

text: volume

text: framework.

text: The

text: re-

text: sulting

text: scheme

text: is

text: of

text: Godunov-type,

text: and

text: an

text: second

text: order

text: ρ

text: z

text: ρ

text: z

text: ρ

text: ρe

text: z

text: ρ

text: e

text: z

text: ρ

text: e

text: a

text: a

text: b

text: b

text: a

text: a

text: a

text: b

text: b

text: b

text: accurate

text: extension

text: is

text: made

text: by

text: using

text: the

text: space-time

text: cou-

text: pled

text: generalized

text: Riemann

text: problem

text: (GRP)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 (a) (b) FIG. 10. Numerical shadow-graph image of the shock-R22 bubble interaction with M = 1.22 by ES-GRP at the exper- s imental time 318µs. M = 1.22 s (c) M = 3 s (d) (e) M = 6 s (f) FIG. 9. Numerical shadow-graph images of the shock-R22 bubble interaction with M = 1.22 obtained by ES-Godunov s FIG. 11. Density contours for the shock-helium bubble inter- (left) and ES-GRP (right) at experimental times (µs): (a)55, action by ES-GRP at t/t = 7.8 (compare with Ref. 40). (b)135, (c)187, (d)247, (e)342 and (f)1020. The correspond- 0 ing experimental shadow-photographs can be found in Ref. 39 (FIGURE 11). pairs. This bridge elongates with time, eventually breaks up. Similarly in FIG. 12(b), two vortices in opposite direc- tions form. The two inner vortices are weaker than the VI. CONCLUSIONS outer vortices because the density gradients are smaller due to diﬀusion between the cylinders. In FIG. 12(c), the inner vortices of the right two cylinders are weaker The study of compressible multi-ﬂuid ﬂows is an im- and the inner gas is pushed upstream. In FIG. 12(d), portant topic in theory, numerics and applications, which the gas in the left cylinder stretches in the span-wise di- can be seen from the very incomplete references quoted rection, forming a bridge between the two outer vortex here. The researches were carried out in various ways ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

label: paragraph

text: (a)

text: (b)

text: FIG.

text: Numerical

label: caption

text: shadow-graph

text: image

text: of

text: the

text: bubble

text: interaction

text: with

text: M

text: by

text: ES-GRP

text: at

text: the

text: exper-

text: s

text: imental

text: time

text: M

text: s

text: (c)

text: M

text: s

text: (d)

text: (e)

text: M

text: s

text: (f)

text: FIG.

text: Numerical

text: shadow-graph

text: images

text: of

text: the

text: bubble

text: interaction

text: with

text: M

text: obtained

text: by

text: ES-Godunov

text: s

text: FIG.

text: Density

text: contours

text: for

text: the

text: shock-helium

text: bubble

text: inter-

text: (left)

text: and

text: ES-GRP

text: (right)

text: at

text: experimental

text: times

text: (µs):

text: action

text: by

text: ES-GRP

text: at

text: t/t

text: (compare

text: with

text: Ref.

text: and

text: The

text: correspond-

text: ing

text: experimental

text: shadow-photographs

text: can

text: be

text: found

text: in

text: Ref.

text: (FIGURE

text: pairs.

text: This

text: bridge

text: elongates

text: with

text: time,

text: eventually

text: breaks

text: up.

text: Similarly

text: in

text: FIG.

text: two

text: vortices

text: in

text: opposite

text: direc-

text: tions

text: form.

text: The

text: two

text: inner

text: vortices

text: are

text: weaker

text: than

text: the

text: VI.

text: CONCLUSIONS

text: outer

text: vortices

text: because

text: the

text: density

text: gradients

text: are

text: smaller

text: due

text: to

text: diﬀusion

text: between

text: the

text: cylinders.

text: In

text: FIG.

text: the

text: inner

text: vortices

text: of

text: the

text: right

text: two

text: cylinders

text: are

text: weaker

text: The

text: study

text: of

text: compressible

text: multi-ﬂuid

text: ﬂows

text: is

text: an

text: im-

text: and

text: the

text: inner

text: gas

text: is

text: pushed

text: upstream.

text: In

text: FIG.

text: portant

text: topic

text: in

text: theory,

text: numerics

text: and

text: applications,

text: which

text: the

text: gas

text: in

text: the

text: left

text: cylinder

text: stretches

text: in

text: the

text: span-wise

text: di-

text: can

text: be

text: seen

text: from

text: the

text: very

text: incomplete

text: references

text: quoted

text: rection,

text: forming

text: a

text: bridge

text: between

text: the

text: two

text: outer

text: vortex

text: here.

text: The

text: researches

text: were

text: carried

text: out

text: in

text: various

text: ways

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTFigure##

text: ##LTFigure##

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text: ##LTFigure##

text: ##LTFigure##


text: 3 In light of the fact ∂ ∇ · (ρφ ) + (ρφ u) = 0. (13) z + z = 1, ρφ = ρ z , (5) ∂t a a a b i i i Therefore, γ is representable for the mixture through the mixture rule is written as the solution of this one-phase mass conservation equa- tion. This, together with the Euler equations (1)-(3), gives a four-equation model in conservation form. It φ + φ = 1, φ e + φ e = e. (6) a b a a b b is typical to use a conservative shock-capturing scheme, such as the Godunov scheme, to numerically solve this model. Let us simulate a material interface separating In this paper, we assume that ﬂuids are modeled by two ﬂuids with diﬀerent temperatures and ratios of spe- the EOS for ideal gases ciﬁc heats. In the Eulerian framework, a moving material interface enters the interior of some computational cells. − As shown in FIG. 1, there is a cell ﬁlled with two separate p = (γ 1)ρ e . (7) k k k k Under Dalton’s law of partial pressures Fluid a Fluid b p = z p + z p , (8) a a b b FIG. 1. Two separate ﬂuids in a cell with diﬀerent tempera- tures the mixture EOS has the uniﬁed form ﬂuids a and b with the same velocity, satisfying the ini- − (cid:54) p = (γ 1)ρe, (9) tial conditions p0a = p0b = p0 and Ta0 = Tb0. For the time being, we assume all that ﬂuid variables on both sides of the material interface in the cell are constants. Re- where γ is the eﬀective ratio of speciﬁc heats for the sulting from the isothermal equilibrium, internal energy mixture given by must exchange between these two ﬂuids. It turns out that pressure p0 changes to p with ﬁxed mass fraction k k φ and total internal energy e. The isothermal equilib- k φ e γ + φ e γ γ = γ(φ , e , e ) = a a a b b b . (10) rium is expressed as a a b φ e + φ e a a b b p p −a = T = T = −b . ρ (γ 1)C a b ρ (γ 1)C So far, various methods computing γ had been pro- a a v,a b b v,b posed in literatures, e.g. Refs. 1 and 22, depending on Conservation of total internal energy before and after diﬀerent model assumptions. In the following, we sketch isothermal equilibrium shows that two typical cases. A. Mass fraction model with isothermal hypothesis φ p φ p a −a + b −b = e = φ C T0 + φ C T0. ρ (γ 1) ρ (γ 1) a v,a a b v,b b a a b b For the ideal gas k, C is assumed to depend only on v,k temperature, and the thermal EOS is Then pa, pb are known. In the end, the pressure p in the cell after isothermal equilibrium described by Dalton’s law e = C T . (11) k v,k k ρφ ρφ p = a p + b p ρ a ρ b a b Following Refs. 10–12, and 16, an isothermal hypoth- − ρφ φ C C − − =p0 a b v,a v,b (γ γ )(T0 T0) esis Ta = Tb is used to express γ in Eq. (10) explicitly φaCv,a + φbCv,b a b a b by is not equal to p0. It turns out that a dramatic change of the pressure arises from the material interface of the φ C γ + φ C γ γ = γ(φ ) = a v,a a b v,b b . (12) ﬂuid mixture. To prevent the pressure error, some a φ C + φ C a v,a b v,b shock-capturing schemes using energy corrections was designed for a UPV ﬂow10 or the convection of the in- The equation of mass conservation for ﬂuid a is ternal energy16. These schemes can reduce the pressure ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: light

text: of

text: the

text: fact

label: equation

text: (ρφ

text: (ρφ

text: u)

text: z

text: z

text: ρφ

text: ρ

text: z

text: ∂t

text: a

text: a

text: a

text: b

text: i

text: i

text: i

text: Therefore,

text: γ

text: is

text: representable

text: for

text: the

text: mixture

text: through

text: the

text: mixture

text: rule

text: is

text: written

text: as

text: the

text: solution

text: of

text: this

text: one-phase

text: mass

text: conservation

text: equa-

text: tion.

text: This,

text: together

text: with

text: the

text: Euler

text: equations

text: gives

text: a

text: four-equation

text: model

text: in

text: conservation

text: form.

text: It

text: φ

text: φ

text: φ

text: e

text: φ

text: e

text: e.

text: a

text: b

text: a

text: a

text: b

text: b

text: is

text: typical

text: to

text: use

text: a

text: conservative

text: shock-capturing

text: scheme,

text: such

text: as

text: the

text: Godunov

text: scheme,

text: to

text: numerically

text: solve

text: this

text: model.

text: Let

text: us

text: simulate

text: a

text: material

text: interface

text: separating

text: In

text: this

text: paper,

text: we

text: assume

text: that

text: ﬂuids

text: are

text: modeled

text: by

text: two

text: ﬂuids

text: with

text: diﬀerent

text: temperatures

text: and

text: ratios

text: of

text: spe-

text: the

text: EOS

text: for

text: ideal

text: gases

text: ciﬁc

text: heats.

text: In

text: the

text: Eulerian

text: framework,

text: a

text: moving

text: material

text: interface

text: enters

text: the

text: interior

text: of

text: some

text: computational

text: cells.

text: As

text: shown

text: in

text: FIG.

text: there

text: is

text: a

text: cell

text: ﬁlled

text: with

text: two

text: separate

text: p

text: (γ

text: e

text: k

text: k

text: k

text: k

text: Under

text: Dalton’s

text: law

text: of

text: partial

text: pressures

text: Fluid

text: a

text: Fluid

text: b

text: p

text: z

text: p

text: z

text: p

text: a

text: a

text: b

text: b

text: FIG.

text: Two

text: separate

text: ﬂuids

text: in

text: a

text: cell

text: with

text: diﬀerent

text: tempera-

text: tures

text: the

text: mixture

text: EOS

text: has

text: the

text: uniﬁed

text: form

text: ﬂuids

text: a

text: and

text: b

text: with

text: the

text: same

text: velocity,

text: satisfying

text: the

text: ini-

text: p

text: (γ

text: tial

text: conditions

text: and

text: For

text: the

text: time

text: being,

text: we

text: assume

text: all

text: that

text: ﬂuid

text: variables

text: on

text: both

text: sides

text: of

text: the

text: material

text: interface

text: in

text: the

text: cell

text: are

text: constants.

text: Re-

text: where

text: γ

text: is

text: the

text: eﬀective

text: ratio

text: of

text: speciﬁc

text: heats

text: for

text: the

text: sulting

text: from

text: the

text: isothermal

text: equilibrium,

text: internal

text: energy

text: mixture

text: given

text: by

text: must

text: exchange

text: between

text: these

text: two

text: ﬂuids.

text: It

text: turns

text: out

text: that

text: pressure

text: changes

text: to

text: p

text: with

text: ﬁxed

text: mass

text: fraction

text: k

text: k

text: φ

text: and

text: total

text: internal

text: energy

text: e.

text: The

text: isothermal

text: equilib-

text: k

text: φ

text: e

text: γ

text: φ

text: e

text: γ

text: γ

text: γ(φ

text: e

text: e

text: a

text: a

text: a

text: b

text: b

text: b

text: rium

text: is

text: expressed

text: as

text: a

text: a

text: b

text: φ

text: e

text: φ

text: e

text: a

text: a

text: b

text: b

text: p

text: p

text: −a

text: T

text: T

text: −b

text: ρ

text: (γ

text: a

text: b

text: ρ

text: (γ

text: So

text: far,

text: various

text: methods

text: computing

text: γ

text: had

text: been

text: pro-

text: a

text: a

text: v,a

text: b

text: b

text: v,b

text: posed

text: in

text: literatures,

text: e.g.

text: Refs.

text: and

text: depending

text: on

text: Conservation

text: of

text: total

text: internal

text: energy

text: before

text: and

text: after

text: diﬀerent

text: model

text: assumptions.

text: In

text: the

text: following,

text: we

text: sketch

text: isothermal

text: equilibrium

text: shows

text: that

text: two

text: typical

text: cases.

text: A.

text: Mass

text: fraction

text: model

text: with

text: isothermal

text: hypothesis

text: φ

text: p

text: φ

text: p

text: a

text: −a

text: b

text: −b

text: e

text: φ

text: C

text: φ

text: C

text: ρ

text: (γ

text: ρ

text: (γ

text: a

text: v,a

text: a

text: b

text: v,b

text: b

text: a

text: a

text: b

text: b

text: For

text: the

text: ideal

text: gas

text: k,

text: C

text: is

text: assumed

text: to

text: depend

text: only

text: on

text: v,k

text: temperature,

text: and

text: the

text: thermal

text: EOS

text: is

text: Then

text: pa,

text: pb

text: are

text: known.

text: In

text: the

text: end,

text: the

text: pressure

text: p

text: in

text: the

text: cell

text: after

text: isothermal

text: equilibrium

text: described

text: by

text: Dalton’s

text: law

text: e

text: C

text: T

text: k

text: v,k

text: k

text: ρφ

text: ρφ

text: p

text: a

text: p

text: b

text: p

text: ρ

text: a

text: ρ

text: b

text: a

text: b

text: Following

text: Refs.

text: and

text: an

text: isothermal

text: hypoth-

text: ρφ

text: φ

text: C

text: C

text: a

text: b

text: v,a

text: v,b

text: (γ

text: γ

text: esis

text: Ta

text: Tb

text: is

text: used

text: to

text: express

text: γ

text: in

text: Eq.

text: explicitly

text: φaCv,a

text: φbCv,b

text: a

text: b

text: a

text: b

text: by

text: is

text: not

text: equal

text: to

text: It

text: turns

text: out

text: that

text: a

text: dramatic

text: change

text: of

text: the

text: pressure

text: arises

text: from

text: the

text: material

text: interface

text: of

text: the

text: φ

text: C

text: γ

text: φ

text: C

text: γ

text: γ

text: γ(φ

text: a

text: v,a

text: a

text: b

text: v,b

text: b

text: ﬂuid

text: mixture.

text: To

text: prevent

text: the

text: pressure

text: error,

text: some

text: a

text: φ

text: C

text: φ

text: C

text: a

text: v,a

text: b

text: v,b

text: shock-capturing

text: schemes

text: using

text: energy

text: corrections

text: was

text: designed

text: for

text: a

text: UPV

text: or

text: the

text: convection

text: of

text: the

text: in-

text: The

text: equation

text: of

text: mass

text: conservation

text: for

text: ﬂuid

text: a

text: is

text: ternal

text: These

text: schemes

text: can

text: reduce

text: the

text: pressure

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 9 Using the same approach as in ES-Godunov to compute This shows that no pressure oscillation appears at the the internal energy for ﬂuid a, we obtain a second-order material interface. non-oscillatory scheme for two-dimensional multi-ﬂuid Finally, we would like to remark that the GRP solver ﬂows. This method has the non-oscillatory property. has the feature that the thermodynamics is deeply char- acterized and embedded into the scheme34. Remark. For high-order Godunov-type methods, the V. NUMERICAL RESULTS energy modiﬁcation can be made to the averaged ﬂow quantities, in the same way as the corresponding ﬁrst or- We present some numerical results by using the current der schemes. Then we implement the linear reconstruc- energy-splitting Godunov-type methods in Section IV . tion process to obtain the linear distribution as in (36). The results are compared with those computed by the Indeed, we consider the solution of the GRP scheme Godunov-type methods in Ref. 10. We abbreviate Is- at an interface with uniform velocity u and pressure p. Godunov for the results by using the Godunov method Using the fact with the isothermal hypothesis11,12, UPV-Godunov for (cid:88) the Godunov results with the energy correction based on a UPV ﬂow10, in addition to the abbreviations: ES- 4 Godunov and ES-GRP. The process of the kinetic energy Λin = 0, (45) j j exchange is added into ES-Godunov and ES-GRP for the j=1 simulation. The following examples show that nonphys- (cid:18) (cid:19) ical oscillations arising from the interface are avoided by and full-discrete scheme (cid:88)(35), we have using the current non-oscillatory conservative schemes. Through the comparison with the corresponding physical (ρφ e )n+1 = (ρφ e )n − 4 Λi zn+12 p− u · n , experimental results, the numerical results show that the a a i a a i j a,i,j γa 1 j current schemes perform well for two-dimensional cases j=1   with very sharp interfaces. For all examples, the CFL and (cid:88)   number is taken to be 0.45. A. Demonstration for the kinetic energy exchange − 4 p · (ρe)n+1 = (ρe)n Λi u n . i i j=1 j γin,j+12 − 1 j The exchange of kinetic energy in the process of ﬂuid mixture has not been well studied in literatures. Hence In view of Eq. (34), we have we propose an example to show the inﬂuence of the ki- netic energy exchange on the distribution of ﬂuids inside mixed cells. We consider an inward two-ﬂuid compression pn+1 zn+1 i− = (ρφ e )n+1 (46) problem, for which the initial discontinuity at x = 0.12 a,i γa 1 a a i separates ﬂuid a with γa = 1.4 in the left from ﬂuid b with γ = 3.0 in the right. These two ﬂuids can be re- b and then garded as air in the left and wolfram in the right. The initial data in the entire computational domain [0, 0.15], composed of 250 cells, are given as pn+1 − zn+1 i− = (ρe)n+1 (ρφ e )n+1, (47) b,i γb 1 i a a i (ρ, u, p, φa) = (0.00129,− 0 , 1.01325, 1), x < 0.12, (ρ, u, p, φ ) = ( 19.237 , 200,1.01325, 1), x > 0.12. b due to the pressure equilibrium.(cid:18)Then we proceed(cid:19)to (cid:88) obtain The left boundary is a solid wall and the right boundary has an inﬂow condition. This problem has exceedingly zn+1 pni−+1 = zn p− − 4 Λi zn+12 p− u · n , huge density ratio and velocity gradient. We use NO-KE a,i γ 1 a,i γ 1 j a,i,j γ 1 j to represent no kinetic energy exchange in the scheme, a a a j=1 (cid:88) (cid:18) (cid:19) and list the numerical results of za in the 199-th cell at and time steps 1 to 5 in Table I. It is observed that without the process of kinetic energy exchange, the volume frac- zn+1 pni−+1 = zn p− − 4 Λi zn+12 p− u · n . tion of air at the interface becomes negative value, which b,i γ 1 b,i γ 1 j b,i,j γ 1 j immediately ruins the numerical simulation. This shows b b b j=1 the necessity of the numerical correction of the kinetic Finally, we can get the pressure at time t energy exchange into the current method. n+1 (cid:88) B. Two-ﬂuid shock-tube problem pn+1 =zn+1pn+1 + zn+1pn+1 i a,i i b,i i − 4 · We consider a two-ﬂuid shock-tube problem in Ref. 7. =p Λij(pu nj) = p. The discontinuity initially at x = 0.3 separates air with j=1 γ = 1.4, C = 0.72 in the left from helium with γ = a v,a b ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Using

text: the

text: same

text: approach

text: as

text: in

text: ES-Godunov

text: to

text: compute

text: This

text: shows

text: that

text: no

text: pressure

text: oscillation

text: appears

text: at

text: the

text: the

text: internal

text: energy

text: for

text: ﬂuid

text: a,

text: we

text: obtain

text: a

text: second-order

text: material

text: interface.

text: non-oscillatory

text: scheme

text: for

text: two-dimensional

text: multi-ﬂuid

text: Finally,

text: we

text: would

text: like

text: to

text: remark

text: that

text: the

text: GRP

text: solver

text: ﬂows.

text: This

text: method

text: has

text: the

text: non-oscillatory

text: property.

text: has

text: the

text: feature

text: that

text: the

text: thermodynamics

text: is

text: deeply

text: char-

text: acterized

text: and

text: embedded

text: into

text: the

text: Remark.

text: For

text: high-order

text: Godunov-type

text: methods,

text: the

text: V.

label: section

text: NUMERICAL

text: RESULTS

text: energy

text: modiﬁcation

text: can

text: be

text: made

text: to

text: the

text: averaged

text: ﬂow

text: quantities,

text: in

text: the

text: same

text: way

text: as

text: the

text: corresponding

text: ﬁrst

text: or-

text: We

text: present

text: some

text: numerical

text: results

text: by

text: using

text: the

text: current

text: der

text: schemes.

text: Then

text: we

text: implement

text: the

text: linear

text: reconstruc-

text: energy-splitting

text: Godunov-type

text: methods

text: in

text: Section

text: IV

text: tion

text: process

text: to

text: obtain

text: the

text: linear

text: distribution

text: as

text: in

text: The

text: results

text: are

text: compared

text: with

text: those

text: computed

text: by

text: the

text: Indeed,

text: we

text: consider

text: the

text: solution

text: of

text: the

text: GRP

text: scheme

text: Godunov-type

text: methods

text: in

text: Ref.

text: We

text: abbreviate

text: Is-

text: at

text: an

text: interface

text: with

text: uniform

text: velocity

text: u

text: and

text: pressure

text: p.

text: Godunov

text: for

text: the

text: results

text: by

text: using

text: the

text: Godunov

text: method

text: Using

text: the

text: fact

text: with

text: the

text: isothermal

text: UPV-Godunov

text: for

text: the

text: Godunov

text: results

text: with

text: the

text: energy

text: correction

text: based

text: on

text: a

text: UPV

text: in

text: addition

text: to

text: the

text: abbreviations:

text: ES-

text: Godunov

text: and

text: ES-GRP.

text: The

text: process

text: of

text: the

text: kinetic

text: energy

text: Λin

text: j

text: j

text: exchange

text: is

text: added

text: into

text: ES-Godunov

text: and

text: ES-GRP

text: for

text: the

text: simulation.

text: The

text: following

text: examples

text: show

text: that

text: nonphys-

text: ical

text: oscillations

text: arising

text: from

text: the

text: interface

text: are

text: avoided

text: by

text: and

text: full-discrete

text: scheme

text: we

text: have

text: using

text: the

text: current

text: non-oscillatory

text: conservative

text: schemes.

text: Through

text: the

text: comparison

text: with

text: the

text: corresponding

text: physical

text: (ρφ

text: e

text: (ρφ

text: e

text: )n

text: Λi

text: p−

text: u

text: n

text: experimental

text: results,

text: the

text: numerical

text: results

text: show

text: that

text: the

text: a

text: a

text: i

text: a

text: a

text: i

text: j

text: a,i,j

text: γa

text: j

text: current

text: schemes

text: perform

text: well

text: for

text: two-dimensional

text: cases

text: with

text: very

text: sharp

text: interfaces.

text: For

text: all

text: examples,

text: the

text: CFL

text: and

text: number

text: is

text: taken

text: to

text: be

text: A.

text: Demonstration

text: for

text: the

text: kinetic

text: energy

text: exchange

text: p

text: (ρe)n

text: Λi

text: u

text: n

text: i

text: i

text: j

text: j

text: The

text: exchange

text: of

text: kinetic

text: energy

text: in

text: the

text: process

text: of

text: ﬂuid

text: mixture

text: has

text: not

text: been

text: well

text: studied

text: in

text: literatures.

text: Hence

text: In

text: view

text: of

text: Eq.

text: we

text: have

text: we

text: propose

text: an

text: example

text: to

text: show

text: the

text: inﬂuence

text: of

text: the

text: ki-

text: netic

text: energy

text: exchange

text: on

text: the

text: distribution

text: of

text: ﬂuids

text: inside

text: mixed

text: cells.

text: We

text: consider

text: an

text: inward

text: two-ﬂuid

text: compression

text: i−

text: (ρφ

text: e

text: problem,

text: for

text: which

text: the

text: initial

text: discontinuity

text: at

text: x

text: a,i

text: γa

text: a

text: a

text: i

text: separates

text: ﬂuid

text: a

text: with

text: γa

text: in

text: the

text: left

text: from

text: ﬂuid

text: b

text: with

text: γ

text: in

text: the

text: right.

text: These

text: two

text: ﬂuids

text: can

text: be

text: re-

text: b

text: and

text: then

text: garded

text: as

text: air

text: in

text: the

text: left

text: and

text: wolfram

text: in

text: the

text: right.

text: The

text: initial

text: data

text: in

text: the

text: entire

text: computational

text: domain

text: composed

text: of

text: cells,

text: are

text: given

text: as

text: i−

text: (ρφ

text: e

text: b,i

text: γb

text: i

text: a

text: a

text: i

text: (ρ,

text: u,

text: p,

text: φa)

text: x

text: (ρ,

text: u,

text: p,

text: φ

text: x

text: b

text: due

text: to

text: the

text: pressure

text: we

text: obtain

text: The

text: left

text: boundary

text: is

text: a

text: solid

text: wall

text: and

text: the

text: right

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text: Figure 6. By choosing an appropriate ﬁtting function we get the desired analytical model denoted as t − . pHL A Note: A visual examination of a ﬁtted curve/plane which is displayed in the Curve Fitting App of Matlab should always be the ﬁrst step in the evaluation of the ﬁtting quality. Beyond the toolbox generates a Goodness-of-Fit Output, which provides goodness-of-ﬁt statistics like the sum of squares due to error (SSE) or the root mean square error (RMSE) [11], [14]. Fig. 7. Comparison between simulated and analytical drain saturation currents − · e.g.: l = (90nm 10nm) rand(5, 1) + 80nm. All values var are within the lists as speciﬁed in Section V.A. Also we use 5 different load capacitances in the interval [10, 200]fF, namely (13, 37, 83, 123, 185)fF. For each parameter combination and load capacitance we compute the analytical inverter delay using the ﬁtting function t − . Overall we get 56 = 15625 values. In order to Fig. 6. Piecewise linear interpolant ﬁtting function tpHL−A which ﬁts tpHL apvHoiLd sAimulating each case with SPICE, we choose 10 dif- dependent from IDnSat−A and CL ferent parameter combinations. For each combination SPICE reports the delay for all ﬁve load capacitances. In Table III V. EXPERIMENTAL RESULTS it is illustrated that the maximum error between SPICE and A. Validation of drain current model analytical model is about 6%. The average error for all cases We start with specifying a list of values (valid for is less than 2%. a 90nm technology node) for each parameter as de- C. Comparison to other analytical models scribed in Section III: L = (90, 88, 86, 84, 82)nm, W = Finally we want to evaluate the quality of recently published (1, 2, 3, 4, 5)µm, TOXE = (3, 2.5, 2, 1.6)nm, V TH0 = inverter delay equations. As reference we use the simulated (0.4, 0.35, 0.3, 0.25)V and U0 = (550, 540, 530, 520)cm2 . SPICE delays for testcase No.1 and the loads C as above. V s L Then 22 SPICE simulations are needed to get the drain Both the Sakurai-Newton (SN) delay metric and the Taur- saturation currents. Together with one additional simulation Ning (TN) delay metric from [8] underestimate the delay for the reference drain current I − (using the reference compared to SPICE. This underestimation becomes smaller DnSat R values for the parameters according to Table ) Matlab’s Curve with increasing loads. The delay T from eq.(6) in [13] also pHL Fitting Toolbox can be feeded. It turns out that for each underestimates the inverter delay, however the underestimation parameter a second order polynomial is an appropriate ﬁt with becomes smaller with decreasing loads. This is illustrated a RMSE below 0.01% in any case. Computing I − in Figure 8. One can argue that in all cases this enormous DnSat A with eq.(5) with all possible combinations of the parameter discrepancy is caused by the mismatch of model and process values listed above leads to 1600 different values. Randomly parameters. But on the other hand a simple additional ﬁtting we choose a subset of 25 testcases. Figure 7 illustrates the factor would not solve the problem due to its dependency on quite good accordance of simulated and calculated saturation C . L drain currents. The maximum error is smaller than 7%, the VI. CONCLUSION average error is 3.0%. In this paper we propose a novel analytical CMOS inverter B. Validation of inverter delay model delay model. The main advantage of this approach compared First we take the parameter settings for the mentioned 25 to previous related work is the more detailled study of the testcases and obtain 5 inverter delays t due to C = drain current especially the transconductance parameter de- pHL L (10, 20, 50, 100, 200)fF for each testcase by SPICE. Together pendent from process parameters. The model generation is with the values of I − for these testcases, Matlab’s both simple because of curve ﬁtting and - apart from some DnSat A Curve Fitting Toolbox generates a piecewise linear interpolant necessary SPICE simulations - computationally inexpensive. ﬁt t − with 125 coefﬁcients. Then we randomly generate Furthermore our method is applicable to any technology node. pHL A 5 values for each parameter with the Matlab routine rand, Experimental results for a wide range of different parameter ##LTFigure## ##LTFigure## ##LTLine##

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text: Test No. Process/Model Parameters Simulated Delay (ps) Analytical Delay (ps) Maximum ErrorAverage Error L(nm)W(µm)TOXE(nm)V TH0(V )U0(cm2 ) (%) (%) V s 1 80.8 2.8 1.78 0.27 538 (13.4,21.9,38.1,51.8,73.4) (14.1,22.5,38.1,51.9,72.4) 5.0 1.8 2 80.8 2.8 2.40 0.37 540 (18.6,32.0,57.4,80.0,114.4) (17.9,31.1,55.9,77.5,111.3) 3.9 3.1 3 80.8 1.34 1.78 0.27 542 (19.5,37.4,71.2,100.6,146.4) (20.0,37.8,71.3,100.1,145.4) 2.5 1.0 4 80.8 4.65 1.78 0.27 538 (11.1,16.1,25.8,34.3,47.0) (11.1,16.1,25.7,33.8,46.6) 1.5 0.6 5 85.3 2.8 1.78 0.33 542 (15.2,24.1,42.3,58.0,82.2) (15.1,25.4,43.1,59.1,82.8) 5.1 2.0 6 85.3 1.34 1.62 0.37 542 (22.7,43.0,80.8,114.0,165.5) (22.8,44.0,83.9,117.5,170.5) 3.7 2.5 7 87.8 2.8 1.78 0.37 540 (15.9,26.3,46.2,63.3,89.9) (15.6,26.3,46.1,63.4,89.9) 1.9 0.4 8 87.8 1.34 1.78 0.27 538 (20.2,38.6,73.2,103.5,150.5) (21.1,39.7,74.9,107.0,153.0) 4.3 2.8 9 87.8 4.65 2.26 0.37 542 (14.4,22.5,37.0,49.9,69.6) (13.8,21.2,35.9,48.5,68.2) 6.1 4.0 10 89.3 2.8 2.07 0.30 538 (15.5,26.1,45.6,62.7,89.4) (15.6,26.2,45.9,63.0,89.6) 0.7 0.5 TABLE III COMPARISON BETWEEN SPICE INVERTER DELAY AND ANALYTICAL DELAY MODEL FOR DIFFERENT PROCESS/MODEL PARAMETERS AND LOAD CAPACITANCES of the 2006 confernce on Asia South Paciﬁc design automation, pp. 61-66, Yokohama, Japan, 2006. [9] N. Chandra, A. K. Yati and A.B. Bhattacharyya, Extended-Sakurai- Newton MOSFET Model for Ultra-Deep-Submicrometer CMOS Digital Design, 22nd International Conference on VLSI Design, 2009. [10] W. Zhao and Y. Cao, New generation of predictive technology model for sub-45nm early design exploration, IEEE Transaction on Electron Devices, vol. 53, no. 11, pp. 2816-2823, 2006. Available online at: http://ptm.asu.edu. [11] @ 1994-2017 The MathWorks, Inc., Curve Fitting Toolbox, online available at: https://de.mathworks.com/help/curveﬁt/index.html, 2017. [12] C. Hu, S. Salahuddin, Y. Singh, S. S. Parihar and C. K. Dabhi, BSIM 4.8.1 MOSFET Model, User’s Manual, Department of Electrical Engineering and Computer Sciences, Berkeley 2017. [13] S. Gummalla, A. R. Subramaniam, Y. Cao and C. Chakrabarti, An Analytical Approach to Efﬁcient Circuit Variability Analysis in Scaled CMOS Design, 13th International Symposium on Quality Electronic Design, 2010. [14] P. Huber and E. Ronchetti, Robust statistics, Wiley New York, 2009. Fig. 8. Comparison of delay metrics from [8], [13] and SPICE simulation [15] N. Weste and D. Harris, CMOS VLSI Design 4th edition, Boston: Addison-Wesley, 2010. values and load capacitances have shown that the worst case error is smaller than 7%. It is expected that our model is suitable to consider process variations in future technologies. REFERENCES [1] Y. Taur and T. H. Ning, Fundamentals of Modern VLSI Devices, Cambridge University Press 1998. [2] J. M. Rabaey, A. Chandrakasar and B. Nikolic, Digital Integrated Circuits, Prentice-Hall, 2003. [3] E. Maricau and G. Gielen, Analog IC Reliability in Nanometer CMOS, Analog Circuits and Signal Processing, Springer Science+Business Media New York, 2013. [4] T. Sakurai and A. R. Newton, Alpha-power law MOSFET model and its applications to ’CMOS inverter delay and other formulas, IEEE Journal of Solid State Circuits, vol. 25, no. 2, pp. 584-594, April 1990. [5] T. Sakurai and A. R. Newton, A simple MOSFET model for circuit analysis, IEEE Transactions on Electron Devices, vol. 38, no. 4, pp. 887-894, April 1991. [6] Y. Wang and M. Zwolinski, Analytical transient response and prop- agation delay model for nanoscale CMOS inverter, Proceedings of International Symposium on Circuits and Systems (ISCAS), pp. 2998- 3001, 2009. [7] F. S. Marranghello, A. I. Reis and R. P. Ribas, Improving Analytical Delay Modeling for CMOS Inverters, Journal of Integrated Circuits and Systems, vol. 10, no. 2, pp. 123-134, 2015. [8] A. Ramalingam, S. V. Kodakara, A. Devgan and D. Z. 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text: Test

label: table

text: No.

text: Process/Model

text: Parameters

text: Simulated

text: Delay

text: (ps)

text: Analytical

text: Delay

text: (ps)

text: Maximum

text: ErrorAverage

text: Error

text: L(nm)W(µm)TOXE(nm)V

text: V

text: s

text: TABLE

text: III

text: COMPARISON

text: BETWEEN

text: SPICE

text: INVERTER

text: DELAY

text: AND

text: ANALYTICAL

text: DELAY

text: MODEL

text: FOR

text: DIFFERENT

text: PROCESS/MODEL

text: PARAMETERS

text: AND

text: LOAD

text: CAPACITANCES

text: of

text: the

text: confernce

text: on

text: Asia

text: South

text: Paciﬁc

text: design

text: automation,

text: pp.

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text: A.B.

text: Bhattacharyya,

text: Extended-Sakurai-

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text: MOSFET

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text: Ultra-Deep-Submicrometer

text: CMOS

text: Digital

text: Design,

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text: VLSI

text: Design,

text: W.

text: Zhao

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text: Y.

text: Cao,

text: New

text: generation

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text: technology

text: model

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text: early

text: design

text: exploration,

text: IEEE

text: Transaction

text: on

text: Electron

text: Devices,

text: vol.

text: no.

text: pp.

text: Available

text: online

text: at:

text: The

text: MathWorks,

text: Inc.,

text: Curve

text: Fitting

text: Toolbox,

text: online

text: available

text: at:

text: C.

text: Hu,

text: S.

text: Salahuddin,

text: Y.

text: Singh,

text: S.

text: S.

text: Parihar

text: and

text: C.

text: K.

text: Dabhi,

text: BSIM

text: MOSFET

text: Model,

text: User’s

text: Manual,

text: Department

text: of

text: Electrical

text: Engineering

text: and

text: Computer

text: Sciences,

text: Berkeley

text: S.

text: Gummalla,

text: A.

text: R.

text: Subramaniam,

text: Y.

text: Cao

text: and

text: C.

text: Chakrabarti,

text: An

text: Analytical

text: Approach

text: to

text: Efﬁcient

text: Circuit

text: Variability

text: Analysis

text: in

text: Scaled

text: CMOS

text: Design,

text: International

text: Symposium

text: on

text: Quality

text: Electronic

text: Design,

text: P.

text: Huber

text: and

text: E.

text: Ronchetti,

text: Robust

text: statistics,

text: Wiley

text: New

text: York,

text: Fig.

text: Comparison

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text: metrics

text: from

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text: SPICE

text: simulation

text: N.

text: Weste

text: and

text: D.

text: Harris,

text: CMOS

text: VLSI

text: Design

text: edition,

text: Boston:

text: Addison-Wesley,

text: values

text: and

text: load

text: capacitances

text: have

text: shown

text: that

text: the

text: worst

text: case

text: error

text: is

text: smaller

text: than

text: It

text: is

text: expected

text: that

text: our

text: model

text: is

text: suitable

text: to

text: consider

text: process

text: variations

text: in

text: future

text: technologies.

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text: Devices,

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text: M.

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text: Sakurai

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text: R.

text: Newton,

text: Alpha-power

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text: MOSFET

text: model

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text: its

text: applications

text: to

text: ’CMOS

text: inverter

text: delay

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text: other

text: formulas,

text: IEEE

text: Journal

text: of

text: Solid

text: State

text: Circuits,

text: vol.

text: no.

text: pp.

text: April

text: T.

text: Sakurai

text: and

text: A.

text: R.

text: Newton,

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text: simple

text: MOSFET

text: model

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text: circuit

text: analysis,

text: IEEE

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text: on

text: Electron

text: Devices,

text: vol.

text: no.

text: pp.

text: April

text: Y.

text: Wang

text: and

text: M.

text: Zwolinski,

text: Analytical

text: transient

text: response

text: and

text: prop-

text: agation

text: delay

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text: (ISCAS),

text: pp.

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text: Marranghello,

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text: I.

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text: and

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text: Improving

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text: Delay

text: Modeling

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text: Inverters,

text: Journal

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text: Integrated

text: Circuits

text: and

text: Systems,

text: vol.

text: no.

text: pp.

text: A.

text: Ramalingam,

text: S.

text: V.

text: Kodakara,

text: A.

text: Devgan

text: and

text: D.

text: Z.

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text: Robust

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text: Delay

text: Modeling

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text: Low

text: Voltage

text: Circuits,

text: Proceedings

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text: MIT-CTP/4975 May 30, 2020 A quantum hydrodynamical description for scrambling and ∗ many-body chaos Mike Blake, Hyunseok Lee, and Hong Liu † ‡ Abstract Recent studies of out-of-time ordered thermal correlation functions (OTOC) in holographic systems and in solvable models such as the Sachdev-Ye-Kitaev (SYK) model have yielded new insights into manifestations of many-body chaos. So far the chaotic behavior has been obtained through explicit calculations in speciﬁc models. In this paper we propose a uniﬁed description of the exponential growth and ballistic butterﬂy spreading of OTOCs across diﬀerent systems using a newly formulated (cid:126) “quantum hydrodynamics,” which is valid at ﬁnite and to all orders in derivatives. The scrambling of a generic few-body operator in a chaotic system is described as building up a “hydrodynamic cloud,” and the exponential growth of the cloud arises from a shift symmetry of the hydrodynamic action. The shift symmetry also shields correlation functions of the energy density and ﬂux, and time ordered correlation functions of generic operators from exponential growth, while leads to chaotic behavior in OTOCs. The theory also predicts an interesting phenomenon of the skipping of a pole at special values of complex frequency and momentum in two-point functions of energy density and ﬂux. This pole-skipping phenomenon may be considered as a “smoking gun” for the hydrodynamic origin of the chaotic mode. ∗ † ‡ 1 ##LTLine##

text: May

text: A

label: title

text: quantum

text: hydrodynamical

text: description

text: for

text: scrambling

text: and

text: many-body

text: chaos

text: Mike

label: author

text: Blake,

text: Hyunseok

text: Lee,

text: and

text: Hong

text: Liu

text: Abstract

text: Recent

label: abstract

text: studies

text: of

text: out-of-time

text: ordered

text: thermal

text: correlation

text: functions

text: (OTOC)

text: in

text: holographic

text: systems

text: and

text: in

text: solvable

text: models

text: such

text: as

text: the

text: Sachdev-Ye-Kitaev

text: (SYK)

text: model

text: have

text: yielded

text: new

text: insights

text: into

text: manifestations

text: of

text: many-body

text: chaos.

text: So

text: far

text: the

text: chaotic

text: behavior

text: has

text: been

text: obtained

text: through

text: explicit

text: calculations

text: in

text: speciﬁc

text: models.

text: In

text: this

text: paper

text: we

text: propose

text: a

text: uniﬁed

text: description

text: of

text: the

text: exponential

text: growth

text: and

text: ballistic

text: butterﬂy

text: spreading

text: of

text: OTOCs

text: across

text: diﬀerent

text: systems

text: using

text: a

text: newly

text: formulated

text: “quantum

text: hydrodynamics,”

text: which

text: is

text: valid

text: at

text: ﬁnite

text: and

text: to

text: all

text: orders

text: in

text: derivatives.

text: The

text: scrambling

text: of

text: a

text: generic

text: few-body

text: operator

text: in

text: a

text: chaotic

text: system

text: is

text: described

text: as

text: building

text: up

text: a

text: “hydrodynamic

text: cloud,”

text: and

text: the

text: exponential

text: growth

text: of

text: the

text: cloud

text: arises

text: from

text: a

text: shift

text: symmetry

text: of

text: the

text: hydrodynamic

text: action.

text: The

text: shift

text: symmetry

text: also

text: shields

text: correlation

text: functions

text: of

text: the

text: energy

text: density

text: and

text: ﬂux,

text: and

text: time

text: ordered

text: correlation

text: functions

text: of

text: generic

text: operators

text: from

text: exponential

text: growth,

text: while

text: leads

text: to

text: chaotic

text: behavior

text: in

text: OTOCs.

text: The

text: theory

text: also

text: predicts

text: an

text: interesting

text: phenomenon

text: of

text: the

text: skipping

text: of

text: a

text: pole

text: at

text: special

text: values

text: of

text: complex

text: frequency

text: and

text: momentum

text: in

text: two-point

text: functions

text: of

text: energy

text: density

text: and

text: ﬂux.

text: This

text: pole-skipping

text: phenomenon

text: may

text: be

text: considered

text: as

text: a

text: “smoking

text: gun”

text: for

text: the

text: hydrodynamic

text: origin

text: of

text: the

text: chaotic

text: mode.

text: ##LTLine##


text: II. QUANTUM HYDRODYNAMICS FOR ENERGY CONSERVATION Consider a quantum many-body system in a “liquid” phase for which the only gapless degrees of freedom are those associated with conserved quantities. The eﬀective theory for these low energy degrees of freedom is hydrodynamics. In this section we discuss quantum hydrodynamics for a system whose only conserved quantity is energy, following the general formulation of [34, 35]. Such a theory applies to systems which have some way to strongly dissipate spatial momenta at some microscopic scales, but at macroscopic level still have spatial translation invariance. With all spatial dependence dropped, the theory describes (0 + 1)-dimensional quantum mechanical systems. Our discussion in this and subsequent sections can be readily generalized to systems with momentum conservations or other global symmetries. A. General quantum hydrodynamics formulation In [34, 35] the hydrodynamics action is formulated as an eﬀective theory for a general statistical system in some state ρ deﬁned on the closed time path (CTP), see left plot of 0 Fig. 7. Let us ﬁrst review the case with full energy-momentum conservation. The formulation is reminiscent of the standard Lagrange description of ﬂuid ﬂows. One introduces a ﬂuid spacetime with coordinates σA = (σ0, σi) where σi can be interpreted labels of each ﬂuid element and σ0 as the “internal clock” of a ﬂuid element. The hydro- dynamical degrees freedom are given by Xµ (σA), which describe motions of ﬂuid elements 1,2 along two segments of the CTP contour,5 and their equations of motion correspond to energy-momentum conservations associated with the two segments. One also introduces an inverse temperature β(σA). Below we will denote the coordinates in physical spacetime as xµ = (t, xi). To construct the hydrodynamic action consider turning on external metrics g and g 1µν 2µν associated with the two legs of the CTP. The hydrodynamic action I is required to hydro 5 A single copy of such kind of variables was already used in [38] and more recently in [22, 39] for ideal ﬂuids. The doubled copies for CTP contour were ﬁrst used in [40] to describe dissipative eﬀects. 12 ##LTLine##

text: II.

text: QUANTUM

text: HYDRODYNAMICS

text: FOR

text: ENERGY

text: CONSERVATION

text: Consider

text: a

text: quantum

text: many-body

text: system

text: in

text: a

text: “liquid”

text: phase

text: for

text: which

text: the

text: only

text: gapless

text: degrees

text: of

text: freedom

text: are

text: those

text: associated

text: with

text: conserved

text: quantities.

text: The

text: eﬀective

text: theory

text: for

text: these

text: low

text: energy

text: degrees

text: of

text: freedom

text: is

text: hydrodynamics.

text: In

text: this

text: section

text: we

text: discuss

text: quantum

text: hydrodynamics

text: for

text: a

text: system

text: whose

text: only

text: conserved

text: quantity

text: is

text: energy,

text: following

text: the

text: general

text: formulation

text: of

text: Such

text: a

text: theory

text: applies

text: to

text: systems

text: which

text: have

text: some

text: way

text: to

text: strongly

text: dissipate

text: spatial

text: momenta

text: at

text: some

text: microscopic

text: scales,

text: but

text: at

text: macroscopic

text: level

text: still

text: have

text: spatial

text: translation

text: invariance.

text: With

text: all

text: spatial

text: dependence

text: dropped,

text: the

text: theory

text: describes

text: quantum

text: mechanical

text: systems.

text: Our

text: discussion

text: in

text: this

text: and

text: subsequent

text: sections

text: can

text: be

text: readily

text: generalized

text: to

text: systems

text: with

text: momentum

text: conservations

text: or

text: other

text: global

text: symmetries.

text: A.

text: General

text: quantum

text: hydrodynamics

text: formulation

text: In

text: the

text: hydrodynamics

text: action

text: is

text: formulated

text: as

text: an

text: eﬀective

text: theory

text: for

text: a

text: general

text: statistical

text: system

text: in

text: some

text: state

text: ρ

text: deﬁned

text: on

text: the

text: closed

text: time

text: path

text: (CTP),

text: see

text: left

text: plot

text: of

text: Fig.

text: Let

text: us

text: ﬁrst

text: review

text: the

text: case

text: with

text: full

text: energy-momentum

text: conservation.

text: The

text: formulation

text: is

text: reminiscent

text: of

text: the

text: standard

text: Lagrange

text: description

text: of

text: ﬂuid

text: ﬂows.

text: One

text: introduces

text: a

text: ﬂuid

text: spacetime

text: with

text: coordinates

text: σA

text: σi)

text: where

text: σi

text: can

text: be

text: interpreted

text: labels

text: of

text: each

text: ﬂuid

text: element

text: and

text: σ0

text: as

text: the

text: “internal

text: clock”

text: of

text: a

text: ﬂuid

text: element.

text: The

text: hydro-

text: dynamical

text: degrees

text: freedom

text: are

text: given

text: by

text: Xµ

text: (σA),

text: which

text: describe

text: motions

text: of

text: ﬂuid

text: elements

text: along

text: two

text: segments

text: of

text: the

text: CTP

text: and

text: their

text: equations

text: of

text: motion

text: correspond

text: to

text: energy-momentum

text: conservations

text: associated

text: with

text: the

text: two

text: segments.

text: One

text: also

text: introduces

text: an

text: inverse

text: temperature

text: β(σA).

text: Below

text: we

text: will

text: denote

text: the

text: coordinates

text: in

text: physical

text: spacetime

text: as

text: xµ

text: (t,

text: xi).

text: To

text: construct

text: the

text: hydrodynamic

text: action

text: consider

text: turning

text: on

text: external

text: metrics

text: g

text: and

text: g

text: 1µν

text: 2µν

text: associated

text: with

text: the

text: two

text: legs

text: of

text: the

text: CTP.

text: The

text: hydrodynamic

text: action

text: I

text: is

text: required

text: to

text: hydro

text: A

text: single

text: copy

text: of

text: such

text: kind

text: of

text: variables

text: was

text: already

text: used

text: in

text: and

text: more

text: recently

text: in

text: for

text: ideal

text: ﬂuids.

text: The

text: doubled

text: copies

text: for

text: CTP

text: contour

text: were

text: ﬁrst

text: used

text: in

text: to

text: describe

text: dissipative

text: eﬀects.

text: ##LTLine##


text: Clearly the equilibrium conﬁguration σ = t, β = β , X = 0 (2.28) 0 a is always a solution to (2.27). For (0 + 1)-dimensional quantum mechanical systems we simply set all the spatial deriva- tives to zero, i.e L − i = H∂ X + M (∂ X )2 + O(a3), (2.29) hydro t a 1 t a 2 and equation of motion reduce to ∂ H = 0, X = 0 . (2.30) t a D. Near-equilibrium quadratic action For our later purpose, let us now consider the quadratic action near equilibrium. Expand- ing around (2.28) we have − − σ = t + (cid:15)(t, xi), β = β + δβ, δβ = β (1 ∂ (cid:15)), X = (cid:15) (t, xi) (2.31) 0 0 t a a and at linear order in δβ we can write H and G as i − − H = f δβ = β f ∂ (cid:15), G = h ∂ δβ = β h ∂ ∂ (cid:15) (2.32) 1 0 1 t i 1 i 0 1 i t where f , h are diﬀerential operators built from ∂ , ∂ . Since X2 terms in (2.24) are al- 1 1 t i a ready quadratic order, thus all β-dependence in M , M should be set to β . With this 1 2 0 understanding, the quadratic action can be written as L − i = (cid:15) K(cid:15) (cid:15) M(cid:15) , K = β (f ∂ + h ∂2)∂ , M = M ∂2 + M ∂2 (2.33) hydro a 2 a a 0 1 t 1 i t 1 t 2 i with equations of motion given by (f ∂ + h ∂2)∂ (cid:15) = 0, (cid:15) = 0 . (2.34) 1 t 1 i t a Imposing dynamical KMS symmetry (2.8) requires that f , h and M satisfy (see Ap- 1 1 1,2 pendix ) − ∗ − iβ ∂ ∗ − iβ ∂ 0 t 0 t β (f f ) = 2i tanh M , β (h + h )∂ = 2i tanh M (2.35) 0 1 1 2 1 0 1 1 t 2 2 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Clearly

text: the

text: equilibrium

text: conﬁguration

text: σ

text: t,

text: β

text: β

text: X

text: a

text: is

text: always

text: a

text: solution

text: to

text: For

text: quantum

text: mechanical

text: systems

text: we

text: simply

text: set

text: all

text: the

text: spatial

text: deriva-

text: tives

text: to

text: zero,

text: i.e

text: L

text: i

text: H∂

text: X

text: M

text: X

text: hydro

text: t

text: a

text: t

text: a

text: and

text: equation

text: of

text: motion

text: reduce

text: to

text: H

text: X

text: t

text: a

text: D.

text: Near-equilibrium

text: quadratic

text: action

text: For

text: our

text: later

text: purpose,

text: let

text: us

text: now

text: consider

text: the

text: quadratic

text: action

text: near

text: equilibrium.

text: Expand-

text: ing

text: around

text: we

text: have

text: σ

text: t

text: xi),

text: β

text: β

text: δβ,

text: δβ

text: β

text: X

text: (t,

text: xi)

text: t

text: a

text: a

text: and

text: at

text: linear

text: order

text: in

text: δβ

text: we

text: can

text: write

text: H

text: and

text: G

text: as

text: i

text: H

text: f

text: δβ

text: β

text: f

text: G

text: h

text: δβ

text: β

text: h

text: t

text: i

text: i

text: i

text: t

text: where

text: f

text: h

text: are

text: diﬀerential

text: operators

text: built

text: from

text: Since

text: terms

text: in

text: are

text: al-

text: t

text: i

text: a

text: ready

text: quadratic

text: order,

text: thus

text: all

text: β-dependence

text: in

text: M

text: M

text: should

text: be

text: set

text: to

text: β

text: With

text: this

text: understanding,

text: the

text: quadratic

text: action

text: can

text: be

text: written

text: as

text: L

text: i

text: K

text: β

text: (f

text: h

text: ∂2)∂

text: M

text: M

text: ∂2

text: M

text: ∂2

text: hydro

text: a

text: a

text: a

text: t

text: i

text: t

text: t

text: i

text: with

text: equations

text: of

text: motion

text: given

text: by

text: (f

text: h

text: ∂2)∂

text: t

text: i

text: t

text: a

text: Imposing

text: dynamical

text: KMS

text: symmetry

text: requires

text: that

text: f

text: h

text: and

text: M

text: satisfy

text: (see

text: Ap-

text: pendix

text: iβ

text: iβ

text: t

text: t

text: β

text: (f

text: f

text: tanh

text: M

text: β

text: (h

text: h

text: tanh

text: M

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: III. HYDRODYNAMIC THEORIES FOR CHAOTIC SYSTEMS The quantum hydrodynamic theory introduced in last section applies to any quantum liquids at a ﬁnite temperature with energy as the only conserved quantity. In this section we show that imposing certain additional symmetry on the action gives rise to exponentially growing behavior of hydrodynamic variable σ. In next section we will show such exponential behavior will not aﬀect correlation functions of the energy density and ﬂux, while in Sec. we show that it leads to chaotic behavior in OTOC. A. Shift symmetry in 0 + 1-dimension To illustrate the general idea, let us ﬁrst consider a quantum mechanical system with no spatial directions. We propose that a chaotic system with Lyapunov exponent λ can be described by an eﬀective theory of the form (2.29) with an additional shift symmetry. More explicitly, let us introduce − − 1 u = e λσ, σ = log u (3.1) λ L and require the Lagrangian to be invariant under a shift symmetry hydro → u u + a (3.2) for arbitrary constant a. Let us look at some immediate implications of this shift symmetry. Recall that equilibrium − state is described by σ = t, which in terms of u, is u = u = e λt. From the shift symmetry, 0 − u = e λt + a must also be a solution to equation of motion (2.30). In terms of β (2.25), this implies that β = β + aβ eλt (3.3) 0 0 is a solution. At the linearized level, this corresponds to an exponentially growing solution in σ(t) − a · · · σ = t eλt + (3.4) λ which we wish to propose as the origin of the chaotic behavior (1.5). 19 ##LTLine## ##LTLine##

text: III.

text: HYDRODYNAMIC

text: THEORIES

text: FOR

text: CHAOTIC

text: SYSTEMS

text: The

text: quantum

text: hydrodynamic

text: theory

text: introduced

text: in

text: last

text: section

text: applies

text: to

text: any

text: quantum

text: liquids

text: at

text: a

text: ﬁnite

text: temperature

text: with

text: energy

text: as

text: the

text: only

text: conserved

text: quantity.

text: In

text: this

text: section

text: we

text: show

text: that

text: imposing

text: certain

text: additional

text: symmetry

text: on

text: the

text: action

text: gives

text: rise

text: to

text: exponentially

text: growing

text: behavior

text: of

text: hydrodynamic

text: variable

text: σ.

text: In

text: next

text: section

text: we

text: will

text: show

text: such

text: exponential

text: behavior

text: will

text: not

text: aﬀect

text: correlation

text: functions

text: of

text: the

text: energy

text: density

text: and

text: ﬂux,

text: while

text: in

text: Sec.

text: we

text: show

text: that

text: it

text: leads

text: to

text: chaotic

text: behavior

text: in

text: OTOC.

text: A.

text: Shift

text: symmetry

text: in

text: To

text: illustrate

text: the

text: general

text: idea,

text: let

text: us

text: ﬁrst

text: consider

text: a

text: quantum

text: mechanical

text: system

text: with

text: no

text: spatial

text: directions.

text: We

text: propose

text: that

text: a

text: chaotic

text: system

text: with

text: Lyapunov

text: exponent

text: λ

text: can

text: be

text: described

text: by

text: an

text: eﬀective

text: theory

text: of

text: the

text: form

text: with

text: an

text: additional

text: shift

text: symmetry.

text: More

text: explicitly,

text: let

text: us

text: introduce

text: u

text: e

text: λσ,

text: σ

text: log

text: u

text: λ

text: L

text: and

text: require

text: the

text: Lagrangian

text: to

text: be

text: invariant

text: under

text: a

text: shift

text: symmetry

text: hydro

text: u

text: u

text: a

text: for

text: arbitrary

text: constant

text: a.

text: Let

text: us

text: look

text: at

text: some

text: immediate

text: implications

text: of

text: this

text: shift

text: symmetry.

text: Recall

text: that

text: equilibrium

text: state

text: is

text: described

text: by

text: σ

text: t,

text: which

text: in

text: terms

text: of

text: u,

text: is

text: u

text: u

text: e

text: λt.

text: From

text: the

text: shift

text: symmetry,

text: u

text: e

text: λt

text: a

text: must

text: also

text: be

text: a

text: solution

text: to

text: equation

text: of

text: motion

text: In

text: terms

text: of

text: β

text: this

text: implies

text: that

text: β

text: β

text: aβ

text: eλt

text: is

text: a

text: solution.

text: At

text: the

text: linearized

text: level,

text: this

text: corresponds

text: to

text: an

text: exponentially

text: growing

text: solution

text: in

text: σ(t)

text: a

text: σ

text: t

text: eλt

text: λ

text: which

text: we

text: wish

text: to

text: propose

text: as

text: the

text: origin

text: of

text: the

text: chaotic

text: behavior

text: ##LTLine##

text: ##LTLine##


text: References 43 I. INTRODUCTION Chaotic phenomena are ubiquitous in nature. While much has been learned about chaos at a classical level, its characterizations and manifestations at a quantum level are far less understood, especially in many-body systems. One characterization of chaos in quantum many-body systems is the Eigenstate Thermal- ization Hypothesis (ETH) [1–5] which says that for a chaotic system, highly excited energy eigenstates should behave like a thermal state. ETH is a powerful statement, implying that chaos underlies thermodynamical behavior of an isolated quantum statistical system. But it is not easy to work with or to check explicitly in practice as energy eigenstates of many- body systems are hard to come by.1 More recently, studies of out-of-time ordered thermal correlation functions in holographic systems and in solvable models like Sachdev-Ye-Kitaev (SYK) model have yielded new insights into manifestations of many-body chaos [7–17]. More explicitly, consider a quantum many-body system at a ﬁnite temperature T = 1 , 0 β0 and the expectation value of the square of the commutator of two generic few-body operators V and W −(cid:10) (cid:11) − C(t) = [V (t), W (0)]2 = C (t) C (t) (1.1) 1 2 (cid:104) (cid:105) (cid:104) (cid:105) C (t) = V (t)W (0)W (0)V (t) + W (0)V (t)V (t)W (0) (1.2) 1 (cid:104) (cid:105) (cid:104) (cid:105) C (t) = V (t)W (0)V (t)W (0) + W (0)V (t)W (0)V (t) (1.3) 2 where C and C are respectively referred to as the time ordered (TOC) and out-of-time 1 2 ordered (OTOC) correlation functions. For convenience throughout the paper we will assume that both V and W are normalized such that they can be considered as dimensionless. When t is small, C(t) should be very small, ∼ 1 C(t) N , tt (1.4) r 1 See also [6] for a recent proposal for another manifestation of quantum chaos in many-body systems. 3 ##LTLine## ##LTLine## ##LTLine##

text: References

text: I.

text: INTRODUCTION

text: Chaotic

text: phenomena

text: are

text: ubiquitous

text: in

text: nature.

text: While

text: much

text: has

text: been

text: learned

text: about

text: chaos

text: at

text: a

text: classical

text: level,

text: its

text: characterizations

text: and

text: manifestations

text: at

text: a

text: quantum

text: level

text: are

text: far

text: less

text: understood,

text: especially

text: in

text: many-body

text: systems.

text: One

text: characterization

text: of

text: chaos

text: in

text: quantum

text: many-body

text: systems

text: is

text: the

text: Eigenstate

text: Thermal-

text: ization

text: Hypothesis

text: (ETH)

text: which

text: says

text: that

text: for

text: a

text: chaotic

text: system,

text: highly

text: excited

text: energy

text: eigenstates

text: should

text: behave

text: like

text: a

text: thermal

text: state.

text: ETH

text: is

text: a

text: powerful

text: statement,

text: implying

text: that

text: chaos

text: underlies

text: thermodynamical

text: behavior

text: of

text: an

text: isolated

text: quantum

text: statistical

text: system.

text: But

text: it

text: is

text: not

text: easy

text: to

text: work

text: with

text: or

text: to

text: check

text: explicitly

text: in

text: practice

text: as

text: energy

text: eigenstates

text: of

text: many-

text: body

text: systems

text: are

text: hard

text: to

text: come

text: More

text: recently,

text: studies

text: of

text: out-of-time

text: ordered

text: thermal

text: correlation

text: functions

text: in

text: holographic

text: systems

text: and

text: in

text: solvable

text: models

text: like

text: Sachdev-Ye-Kitaev

text: (SYK)

text: model

text: have

text: yielded

text: new

text: insights

text: into

text: manifestations

text: of

text: many-body

text: chaos

text: More

text: explicitly,

text: consider

text: a

text: quantum

text: many-body

text: system

text: at

text: a

text: ﬁnite

text: temperature

text: T

text: β0

text: and

text: the

text: expectation

text: value

text: of

text: the

text: square

text: of

text: the

text: commutator

text: of

text: two

text: generic

text: few-body

text: operators

text: V

text: and

text: W

text: C(t)

text: [V

text: (t),

text: W

text: C

text: (t)

text: C

text: (t)

text: C

text: (t)

text: V

text: (t)W

text: (t)

text: W

text: (t)V

text: (t)W

text: C

text: (t)

text: V

text: (t)W

text: (t)W

text: W

text: (t)W

text: (t)

text: where

text: C

text: and

text: C

text: are

text: respectively

text: referred

text: to

text: as

text: the

text: time

text: ordered

text: (TOC)

text: and

text: out-of-time

text: ordered

text: (OTOC)

text: correlation

text: functions.

text: For

text: convenience

text: throughout

text: the

text: paper

text: we

text: will

text: assume

text: that

text: both

text: V

text: and

text: W

text: are

text: normalized

text: such

text: that

text: they

text: can

text: be

text: considered

text: as

text: dimensionless.

text: When

text: t

text: is

text: small,

text: C(t)

text: should

text: be

text: very

text: small,

text: C(t)

text: N

text: tt

text: r

text: See

text: also

text: for

text: a

text: recent

text: proposal

text: for

text: another

text: manifestation

text: of

text: quantum

text: chaos

text: in

text: many-body

text: systems.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##