発表ファイル数理物理・物性基礎論セミナー Mallick 1

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Recent Developments in Non-Equilibrium

Statistical Physics

K. Mallick

Institut de Physique Th´eorique, CEA Saclay (France)

Tokyo Institute of Technology, March 30, 2016

K. Mallick Recent Developments in Non-Equilibrium Statistical Physics

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Introduction

The statistical mechanics of a system at thermal equilibrium is encoded in theBoltzmann-Gibbs canonical law:

Peq(_{C) =} ^e^{−E (C)/kT} Z

thePartition Function Zbeing related to the ThermodynamicFree Energy F:

F =_{−kTLog Z}

This provides us with awell-defined prescriptionto analyze systems at equilibrium:

(i) Observables are mean values w.r.t. thecanonical measure.

(ii) Statistical Mechanics predictsfluctuations(typically Gaussian) that are out of reach of Classical Thermodynamics (Brownian Motion).

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Systems far from equilibrium

Consider a Stationary Driven System in contact with reservoirs at different potentials: no microscopic theory is yet available.

R1

J

R2

• What are therelevant macroscopic parameters?

• Which^functionsdescribe the state of a system?

• Do Universal Lawsexist? Can one define Universality Classes?

• Can one postulate a general form for themicroscopic measure?

• What do the fluctuationslook like (‘non-gaussianity’)?

In the steady state, a non-vanishing macroscopic current J flows, thus breaking time-reversal invariance

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EQUILIBRIUM

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Lars Onsager (1903-1976)

‘As in other kinds of book-keeping, the trickiest questions that arise in the application of thermodynamics deal with the proper identification and classification of the entries; the arithmetics is straightforward’ (Onsager, 1967).

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FIRST PRINCIPE

∆ _{U = W + Q}

THE ENERGY OF THE UNIVERSE IS CONSTANT.

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IRREVERSIBILITY

Whenever dissipation and heat exchanges are involved, time reversibility seems to be lost

SOME EVENTS ARE ALLOWED BY NATURE BUT NOT THE OTHERS!

A criterion for separating allowed processes from impossible one is required (Clausius, Kelvin-Planck).

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SECOND PRINCIPLE

A NEW physical concept (Clausius): ENTROPY.

S 2 _{− S} 1 _≥

R

1 _→2

∂Q

T

Clausius Inequality (1851)

THE ENTROPY OF THE UNIVERSE INCREASES.

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The Mistress of the World and Her Shadow

• A system wants tominimize its energy.

• A system wants tomaximize its entropy.

This competition between energy and entropy is at the heart of most of everyday physical phenomena (such as phase transitions: ice_{→ water).}

The two principles of thermodynamics can be embodied simultaneously by theFREE ENERGY F :

F = U_{− TS}

The decrease of free energy represents the maximal work that one can extract from a system.

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Free energy: Maximal Work Theorem

Consider a gas enclosed in a chamber with a moving piston. We suppose that the gas evolves from state A to B and that it can exchange heat only with it environment at fixed temperature T .

V A

V B

T T

Because of irreversibility, theWork,_Wuseful, that one can extract from this system isat most equal toto the decrease of free energy:

hW^usefuli ≤F^initial− F^final⁼−∆F

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STATISTICAL MECHANICS

J. C. Maxwell L. Boltzmann

The connection with thermodynamics is given byBoltzmann’s formulaor, equivalently

F =_{−kTLog Z}

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Thermal Equilibrium: a dynamical state

Equilibrium is a dynamical concept. At the molecular scale things constantly changeand a system keeps on evolving through various microscopic configurations:

Thermodynamic observables are nothing but average values of fluctuating, probabilistic, microscopic quantities.

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Physics of Brownian Motion: The Einstein Formula

The Brownian Particle is restlessly shaken by water molecules. It diffuses asa random-walker.

D = ₆ ^RT

πηa _N

R: Perfect Gas Constant T: Temperature η : viscosity of water a: diameter of the pollen

N : Avogadro Number

Jean Perrin: ‘I have weighted the Hydrogen Atom’

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Fluctuation-Dissipation Relation

Suppose that the Brownian Particle is subject to a small force fext. Balancing with the viscous force_−(6πηa)v (Stokes) gives the limiting speed

v_∞=σfext withσ = ¹ 6πηa Theresponsecoefficientσ is called asusceptibility. The Einstein Relation can be rewritten as

σ = ^D kT

Susceptibility (Linear Response)_≡Fluctuations at Equilibrium (Kubo Formula)

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Time-reversal Invariance and Detailed Balance

The microscopic equations are invariant by time-reversal: the probability of a given trajectory in phase-space is equal to the probability of the time reversed trajectory.

C

n

C₂ C

0

C₁

t

1 ^t2 ^tn

TRAJECTORY C(t)

0 T

Cn

C1

C0

C₂ _T₋ t 1 t T−

2

T₋t

n

TIME−REVERSED TRAJECTORY C(T−t)

T 0

DETAILED BALANCE:

e⁻

E_(C)

kT _W₍_C→C′₎

e⁻

E_(C′)

kT _W₍_C′_→C)

= 1

Onsager (1931)

A system is at thermal equilibrium iff it satisfies detailed-balance.

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Onsager’s Reciprocity Relations (1931)

∆Τ

∆Τ1

2

3

Ji=₋^P³_k=1Lik_∂x^∂T_k

(Fourier Law)

The Conductivity Tensor L remainssymmetriceven though the crystal does not display any special symmetry

Lik = Lki

Crucial for Thermoelectric Effects.

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Linear Response Theory

Brownian Fluctuations show that Equilibrium is a dynamical concept.

The fact that the dynamics converges towardsthermodynamic equilibriumandtime-reversal invariance(detailed-balance) are the key-properties behind Einstein and Onsager’s Relations.

Thermodynamic equilibrium is characterized by the fact that the average values of all thefluxes exchanged between the system and its

environment (matter, charge, energy, spin...) identically vanish.

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OUT OF EQUILIBRIUM

In Nature, many systems are far from thermodynamic equilibrium and keep on exchanging matter, energy, information with their surroundings. There is no general conceptual framework to study such systems.

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A Surprise: The Jarzynski Identity

Remember the maximal work inequality: hW i ≤ F^A− F^B ⁼−∆F

We put brackets to emphasize that we consider theaverage work: Statistical Physics has taught us that physical observables fluctuate.

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A Surprise: The Jarzynski Identity

Remember the maximal work inequality: hW i ≤ F^A− F^B ⁼−∆F

We put brackets to emphasize that we consider theaverage work: Statistical Physics has taught us that physical observables fluctuate.

It was found very recently that there exists aremarkable equalitythat underliesthis classical inequality.

D ekT^W

E= e⁻^∆F^kT

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The Jarzynski Identity

DekT^W

E= e⁻^∆F^kT

Jarzynski’s Work Theorem (1997)

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Consequences

1. Jarzynski’s identitymathematically implies the good old maximal work inequality.

2. But, in order to have an EQUALITY, there must exist some occurrences in which

W >_−∆F

There must be instances in which the classical inequality which results from the Entropy Principle is ‘violated’.

3. Jarzynski’s identity was checked experimentally on single RNA folding/unfolding experiments (Bustamante et al.): it has experimental applications in biophysics and at the nanoscale.

4. The relation of Crooks: a refinement Jarzynski’s identity that allows us to quantify precisely the ‘transient violations of the second principle’.

P^F_{(W )} P^R₍_{−W )} ^{= e}

W −∆F

kT (Crooks,1999)

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Rare Events and Large Deviations

Let ǫ1, . . . , ǫN be N independent binary variables, ǫk =_{±1, with} probability p (resp. q = 1− p). Their sum is denoted by S^N ⁼^P^N1 ^ǫ^k^.

• The Law of Large Numbersimplies that SN/N→ p − q a.s.

• The Central Limit Theoremimplies that [SN− N(p − q)]/^√^N converges towards a Gaussian Law.

One can show that for−1 < r < 1, in the large N limit, Pr^S^N

N ^{= r}

∼ e^{−N Φ(r)}

where the positive function Φ(r ) vanishes for r = (p_{− q).} The function Φ(r ) is aLarge Deviation Function: it encodes the probability of rare events.

Φ(r ) = ^{1 + r} 2 ^ln

1 + r 2p

+¹^{− r}

2 ^ln

1_{− r} 2q

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Local density fluctuations in a gas at thermal

equilibrium

V, T N

v n

Mean Density ρ0= ^N_V In a volume v s. t. 1_{≪ v ≪ V}

hⁿvi = ρ⁰

The local density varies around ρ0. Typical fluctuations scale as pv/V . The probability of observing large fluctuations is given by

Probaⁿ v ^{= ρ}

∼ e^{−v Φ(ρ)} ^{with Φ(ρ}⁰^{) = 0}

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Large Deviations of the Density fluctuations

How can we calculate the Large Deviation FunctionΦ(ρ) using elementary statistical mechanics?

We must count the fraction of the configurations of the gas that have n= ρv particles in the small volumev andN_{− n}particles in the rest of the volumeV _{− v}.

Suppose that the interactions of the gas molecules are local. Then, neglecting surface effects, this number is given by

Probaⁿ v ^{= ρ}

≃ ^Z(v , n, T )Z (V − v, N − n, T ) Z(V , N, T )

Use that, by definition,

Z(v , n, T ) = e−vβf (ρ,T )

where β = 1/kBT is the inverse temperature and f (ρ, T ) is the free energy per unit volume and perform an expansion for 1_{≪ v ≪ V .}

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The Large Deviation Function for density fluctuations is given by Φ(ρ) = β

f(ρ, T )_{− f (ρ}0, T )_{− (ρ − ρ}0)^∂f

∂ρ0

We can ask the more general question of the large deviation of a density profile: cover the large box with K = V /v small boxes and calculate the probability of having a density ρ1in the first box, ρ2in the second box ...

Proba_(ρ₁_{, ρ}₂_{, . . . ρ}_K₎_{≃ e}^{−V F(ρ}¹^,ρ²^,...ρ^K⁾

A reasoning similar to the one above allows us to show that Proba_(ρ₁_{, ρ}₂_{, . . . ρ}_K₎_∼

Q

k^Z⁽ⁿ^k^{, v , T )}

Z(V , N, T ) Taking the infinite volume limit, we obtain

F(ρ¹^{, ρ}²^{, . . . ρ}^K^{) =} _K^β

K

X

k=1

(f (ρi, T )_{− f (ρ}0, T ))

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The Free Energy as a L. D. F.

If we let the number K of boxes go to infinity, then the question we are asking is theprobability of observing a given density profile ρ(x) in the big volume V . The large deviation functionF becomes a functional of the density profile:

F[ρ(x)] = β Z

dx(f (ρ(x), T )_{− f (ρ}0, T ))

f =− log Z (ρ, T ) being, as above, the free energy per unit volume. The Free Energy of Thermodynamics can be viewed as a Large Deviation Function

Conversely, large deviation functions may play the role of potentials in non-equilibrium statistical mechanics. Indeed, they can be defined for very general processes, even far from equilibrium.

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Density Fluctuations

For a gas in a room, at thermal equilibrium, the probability of observing a density profileρ(x)takes the form:

Pr_{{ρ(x)} ∼ e}−βV F({ρ(x)}

F({ρ(x)}) = Z 1

0

(f (ρ(x), T )_{− f (¯}ρ, T )) d³x The Free Energy can be viewed as a Large Deviation Function.

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Density Fluctuations

Pr_{{ρ(x)} ∼ e}−βV F({ρ(x)}

F({ρ(x)}) = Z 1

0

What happens out of equilibrium?

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Density Fluctuations

Pr_{{ρ(x)} ∼ e}−βV F({ρ(x)}

F({ρ(x)}) = Z 1

0

What happens out of equilibrium?

R1 ^R2

What is the probability of observing anatypical density profile in the steady state? What does the functional F({ρ(x)}) look like for such a non-equilibrium system?

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Large Deviations of the Total Current

R1

J

R2

Let Yt be the total charge transported through the system(total current) between time 0 and time t.

In the stationary state: a non-vanishing mean-current ^Y_t^t _{→ J} The fluctuations ofYt obey aLarge Deviation Principle:

P^Y^t t ^{= j}

∼e^−tΦ(j)

Φ(j)being the large deviation functionof the total current.

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The Gallavotti and Cohen Symmetry

Large deviation functions obey remarkable identities that remain valid far from equilibrium: The Fluctuation Theorem of Gallavotti and Cohen. Large deviation functions obey a symmetry that remains valid far from equilibrium:

Φ(j)− Φ(−j) = αj

Equivalently,

Prob( j ) Prob(_−j)^∼e

−tαj

This Fluctuation Theorem of Gallavotti and Cohenis deep and general: it reflects covariance properties undertime-reversal.

In the vicinity of equilibrium the Fluctuation Theorem yields the fluctuation-dissipation relation (Einstein), Onsager’s relations and linear response theory (Kubo).

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The General Large Deviations Problem

More generally, the probability to observe anatypicalcurrent j(x, t) and the corresponding density profile ρ(x, t) during 0_{≤ s ≤ L}²T (L being the size of the system) is given by

Pr{j(x, t), ρ(x, t)} ∼ e^{−L I(j,ρ)}

Is there a Principle which gives this large deviation functional for systems out of equilibrium?

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Why study Large Deviations?

Equilibrium Thermodynamic potentials (Entropy, Free Energy) can be defined as large deviation functions.

Large deviations are well defined far from equilibrium: they are good candidates for being non-equilibrium potentials.

Large deviation functions obey remarkable identities, valid far from equilibrium (Gallavotti-Cohen Fluctuation Theorem; Jarzynski and Crooks Relations).

These identities imply, in the vicinity of equilibrium, the fluctuation dissipation relation (Einstein), Onsager’s relations and linear response theory (Kubo).

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SOME EXACT RESULTS

FAR FROM EQUILIBRIUM

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Study Non-Equilibrium via Model Solving

The fundamental non-equilibrium system

R1

J

R2

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Study Non-Equilibrium via Model Solving

The fundamental non-equilibrium system

R1

J

R2

The asymmetric exclusion model with open boundaries (ASEP)

q ₁

γ _δ

1 L

RESERVOIR RESERVOIR

α _β

Thousands of articles devoted to this model in the last 20 years: Paradigm for non-equilibrium behaviour

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An Elementary Model for Protein Synthesis

C. T. MacDonald, J. H. Gibbs and A.C. Pipkin, Kinetics of biopolymerization on nucleic acid templates, Biopolymers (1968).

=3

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SOME APPLICATIONS

• Interacting Brownian Processes (Spitzer, Harris, Liggett)^.

• Driven diffusive systems (Katz, Lebowitz and Spohn).

• Transport of Macromolecules through thin vessels. Motion of RNA templates.

• Hopping conductivity in solid electrolytes.

• Directed Polymers in random media. Reptation models.

• Interface dynamics. KPZ equation

• Traffic flow.

• Sequence matching.

• Brownian motors.

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The Matrix Ansatz for ASEP (DEHP, 1993)

The stationary probability of a configurationC is given by

P(_{C) =} ¹ ZL^{hW |}

L

Y

i=1

(τiD+ (1_{− τ}i)E)_{|V i}

where τi = 1 (or 0) if the site i is occupied (or empty). The normalization constant ZL=_{hW |}(D+E)^L_{|V i}.

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The Matrix Ansatz for ASEP (DEHP, 1993)

The stationary probability of a configurationC is given by

P(_{C) =} ¹ ZL^{hW |}

L

Y

i=1

(τiD+ (1_{− τ}i)E)_{|V i}

where τi = 1 (or 0) if the site i is occupied (or empty). The normalization constant ZL=_{hW |}(D+E)^L_{|V i}. The operatorsD andE, the vectors_{hW |} and_{|V i} satisfy

DE _{− q}ED = (1_{− q) (}D +E) (βD_{− δ}E)_{|V i} = _{|V i}

hW |^(α^E− γ^D) ⁼ hW |

This algebra, related to q-deformed oscillators, encodes the stationary properties of the system and allows us to derive the exact phase diagram of the model.

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The Phase Diagram of the open ASEP

LOW DENSITY

HIGH DENSITY MAXIMAL CURRENT

ρ 1 − ρ

a b

1/2 1/2

ρa =_a₊¹₊₁ : effective left reservoir density. ρb= _b₊^b₊₁⁺ : effective right reservoir density.

a_±= ⁽¹− q − α + γ) ±p(1 − q − α + γ)²+ 4αγ 2α

b_±=⁽¹− q − β + δ) ±p(1 − q − β + δ)²^{+ 4βδ} 2β

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Large Deviations of the Density Profile in ASEP

The probability of observing anatypical density profile in the steady state of the ASEPwas calculatedstarting from Matrix Ansatz for the exact microscopic solution(B. Derrida, J. Lebowitz E. Speer, 2002). In thesymmetric case q= 1:

F({ρ(x)}) = Z 1

0

dx

B(ρ(x), F (x)) + log ^F^′^(x) ρ2_{− ρ}1

where B(u, v ) = (1_{− u) log}^1−u_1−v + u log^u_v and F (x) satisfies

F F^′2+ (1_{− F )F}^′′ = F^′2ρ with F(0) = ρ1and F (1) = ρ2. This functional is non-local as soon as ρ1_{6= ρ}2.

This functional is NOT identical to the one given by local equilibrium.

Note that in the case of equilibrium, forρ1= ρ2= ¯ρ, we recover F({ρ(x)}) =

Z 1

0

dx

(1− ρ(x)) log¹^{− ρ(x)}₁

− ¯^ρ ^{+ ρ(x) log} ρ(x)

¯ ρ

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Current Statistics

A parametric representation of the cumulant generating function E (µ) is obtainedusing integrability techniques (Bethe Ansatz).

For α = β = 1:

µ = ₋

∞

X

k=1

(2k)! k!

[2k(L + 1)]! [k(L + 1)]! [k(L + 2)]!

B^k 2k ^,

E = ₋

X∞ k=1

(2k)! k!

[2k(L + 1)_{− 2]!} [k(L + 1)− 1]! [k(L + 2) − 1]!

B^k 2k ^. First cumulants of the current

Mean Value : J =_2(2L+1)^L+2 Variance : ∆ = ³₂(4L+1)![L!(L+2)!]²

[(2L+1)!]³(2L+3)!

Skewness :

E3= 12^[(L+1)!](2L+1)[(2L+2)!]²^[(L+2)!]³⁴

n9(L+1)!(L+2)!(4L+2)!(4L+4)!

(2L+1)![(2L+2)!]²[(2L+4)!]² − 20(3L+2)!(3L+6)!^(6L+4)!

o For large systems: E3_→ ^2187−1280

√3

10368 ^π∼ −0.0090978...

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Large Deviation Function of the Current

In the limit of large size systems, the following exact expression is found for the Large Deviation Function of the current:

Φ(j) = (1_{− q)}ⁿρa− r + r(1 − r) ln^1−ρρa^a

r 1−r

o

where the current j is parametrized asj= (1− q)r(1 − r).

-0.03 -0.028 -0.026 -0.024 -0.022 -0.02 -0.018 -0.016 -0.014 -0.012 -0.01

0 10 20 30 40 50 60 70 80 90 100 C3*(L)

L α = 0.50, β = 0.65

DMRG results exact results

-0.009 -0.008 -0.007 -0.006 -0.005 -0.004 -0.003 -0.002

0 10 20 30 40 50 60 70 80 90 100 C3*(L)

L α = 0.65, β = 0.65

DMRG results exact results

SKEWNESS

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The Hydrodynamic Limit: Diffusive case

E =

_ν _/

_2L

ρ _ρ

1 2

L

Starting from the microscopic level, define local density ρ(x, t) and current j(x, t) with macroscopic space-time variables x = i/L, t = s/L² (diffusive scaling).

The typical evolution of the system is given by the hydrodynamic behaviour (Burgers-type equation):

∂tρ =∇ (D(ρ)∇ρ) − ν∇σ(ρ) ^with ^{D(ρ) = 1}^andσ(ρ) = 2ρ(1_{− ρ)} (Lebowitz, Spohn, Varadhan)

How can Fluctuations be taken into account?

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Fluctuating Hydrodynamics

Consider Yt the total number of particles transfered from the left reservoir to the right reservoir during time t.

limt_→∞^hY_t^tⁱ =D(ρ)^ρ¹^−ρ_L ² +σ(ρ)^ν_L for (ρ1_{− ρ}2) small limt_→∞^hY

2 ti

t ⁼

σ(ρ)

L ^{for ρ}¹^{= ρ}²= ρ and ν = 0. Then, the equation of motion is obtained as:

∂tρ =_−∂xj with j=₋D(ρ)_{∇ρ + ν}σ(ρ)+pσ(ρ)ξ(x, t) where ξ(x, t) is a Gaussian white noise with variance

hξ(x^′^{, t}^′^{)ξ(x, t)}i =_L¹^δ(x− x^′^)δ(t− t^′⁾

For the symmetric exclusion process, the ‘phenomenological’ coefficients are given by

D(ρ) = 1 and σ(ρ) = 2ρ(1_{− ρ)}

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A General Principle for Large Deviations?

The probability to observe anatypicalcurrent j(x, t) and the

corresponding density profile ρ(x, t) during a time L²T (L being the size of the system) is given by

Pr{j(x, t), ρ(x, t)} ∼ e^{−L I(j,ρ)}

A general principle has been found(G. Jona-Lasinio et al.), to express this large deviation functional_{I(j, ρ)}as an optimal path problem:

I(j, ρ) = min_ρ,j ⁿ Z T

0

dt Z 1

0

dx^(j− νσ(ρ) + D(ρ)∇ρ)² 2σ(ρ)

o

with theconstraint: ∂tρ =_−∇.j

KnowingI(j, ρ), one could derive the large deviations of the current and of the density profile. For instance,Φ(j) = minρ_{{I(j, ρ)}}

However, at present, the available results for this variational theory are precisely the ones given by exact solutions of the ASEP.

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Macroscopic Fluctuation Theory

Mathematically, one has to solve the corresponding Euler-Lagrange equations. TheHamiltonian structureis expressed by a pair of conjugate variables (p, q).

After some transformations, one obtains a set of coupled PDE’s (here, we take ν = 0):

∂tq = ∂x[D(q)∂xq]_{− ∂}x[σ(q)∂xp]

∂tp = _−D(q)∂xxp₋¹ 2^σ

′_(q)(∂ xp)²

where q(x, t) is the density-field and p(x, t) is a conjugate field. The’transport coefficients’D(q)(= 1)andσ(q)(= 2q(1_{− q))}contain the information of the microscopic dynamics relevant at the macroscopic scale.

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Macroscopic Fluctuation Theory

Mathematically, one has to solve the corresponding Euler-Lagrange equations. TheHamiltonian structureis expressed by a pair of conjugate variables (p, q).

After some transformations, one obtains a set of coupled PDE’s (here, we take ν = 0):

∂tq = ∂x[D(q)∂xq]_{− ∂}x[σ(q)∂xp]

∂tp = _−D(q)∂xxp₋¹ 2^σ

′_(q)(∂ xp)²

where q(x, t) is the density-field and p(x, t) is a conjugate field. The’transport coefficients’D(q)(= 1)andσ(q)(= 2q(1_{− q))}contain the information of the microscopic dynamics relevant at the macroscopic scale.

A general framework but the MFT equations are very difficult to solve in general. By using them one can in principle calculate large deviation functionsdirectly at the macroscopic level.

The analysis of this new set of ‘hydrodynamic equations’ has just begun!

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Conclusions

Non-Equilibrium Statistical Physics has undergone remarkable developments in the last two decades and a unified framework is emerging.

Large deviation functions(LDF) appear as ageneralization of the thermodynamic potentialsfor non-equilibrium systems. They satisfy remarquable identities(Gallavotti-Cohen, Jarzynski-Crooks) valid far from equilibrium.

The LDF’s are very likely to play a key-role in the future of non-equilibrium statistical mechanics.

Current fluctuations are a signature of non-equilibrium behaviour. The exact results derived for the Exclusion Process can be used to calibrate the more general framework ofmacroscopic fluctuation theory (MFT), which is currently being developed.

発表ファイル 数理物理・物性基礎論セミナー Mallick 1

Recent Developments in Non-Equilibrium

Statistical Physics

Introduction

Systems far from equilibrium

EQUILIBRIUM

Lars Onsager (1903-1976)

FIRST PRINCIPE

∆ U = W + Q

IRREVERSIBILITY

SECOND PRINCIPLE

S 2 − S 1 ≥

R

1 →2

∂Q

T

The Mistress of the World and Her Shadow

Free energy: Maximal Work Theorem

T T

STATISTICAL MECHANICS

Thermal Equilibrium: a dynamical state

Physics of Brownian Motion: The Einstein Formula

D = 6 RT

πηa N

Fluctuation-Dissipation Relation

Time-reversal Invariance and Detailed Balance

= 1

Onsager’s Reciprocity Relations (1931)

Linear Response Theory

OUT OF EQUILIBRIUM

A Surprise: The Jarzynski Identity

A Surprise: The Jarzynski Identity

The Jarzynski Identity

Consequences

Rare Events and Large Deviations

Local density fluctuations in a gas at thermal

equilibrium

Large Deviations of the Density fluctuations

The Free Energy as a L. D. F.

Density Fluctuations

Density Fluctuations

Density Fluctuations

Large Deviations of the Total Current

The Gallavotti and Cohen Symmetry

The General Large Deviations Problem

Why study Large Deviations?

SOME EXACT RESULTS

FAR FROM EQUILIBRIUM

Study Non-Equilibrium via Model Solving

Study Non-Equilibrium via Model Solving

An Elementary Model for Protein Synthesis

The Matrix Ansatz for ASEP (DEHP, 1993)

The Matrix Ansatz for ASEP (DEHP, 1993)

The Phase Diagram of the open ASEP

Large Deviations of the Density Profile in ASEP

Current Statistics

Large Deviation Function of the Current

The Hydrodynamic Limit: Diffusive case

ν /

ρ ρ

L

Fluctuating Hydrodynamics

A General Principle for Large Deviations?

Macroscopic Fluctuation Theory

Macroscopic Fluctuation Theory

Conclusions

発表ファイル数理物理・物性基礎論セミナー Mallick 1

∆ _{U = W + Q}

S 2 _{− S} 1 _≥

1 _→2

D = ₆ ^RT

πηa _N

_ν _/

ρ _ρ