New concepts emerging from a linear response theory for - - PowerPoint PPT Presentation

New concepts emerging from a linear response theory for nonequilibrium

Marco Baiesi Physics and Astronomy Department “Galileo Galilei”, University of Padova, and INFN

• in colmaboration with Christian Maes, Urna Basu,

Eugenio Lippielmo, Alessandro Sarracino, Bram Wynants, Eliran Boksenbojm

GGI Workshop, 26-30 May, 2014

Overview

Linear response Linear response to temperature kicks Entropy production and something else Time-symmetric quantities: how many?

A macroscopic FPU?

Livia Conti, Lamberto Rondoni, et al: www.rarenoise.lnl.infn.it

Linear response for FPU?

How does a Fermi-Pasta-Ulam chain react to a change in

• ne temperature?

Nonequilibrium specific heat variance of the energy Compressibility in nonequilibrium?

6=

Fluctuation-Dissipation Th.: Kubo

An observable A(t) reacts to the appearance

• f a potential V(s)

is the entropy production from -V

dhA(t)i dhs = 1 T d dshA(t)V (s)i E → E − hsV

1 T d dsV (s)

Out of equilibrium: many FDT

a1) perturb the density of states and evolve

(Agarwal, Vulpiani & C, Seifert & Speck, Parrondo &C,…)

a2) “bring back” the observable to the perturbation (Ruelle) b) probability of paths (Cugliandolo &C, Harada-

Sasa, Lippiello &C, Ricci-Tersenghi, Chatelain, Maes, …) short review: Baiesi & Maes, New J. Phys. (2013)

Path probability (Markov),

Overdamped Langevin

• Discrete states C, C’, …with jump rates

ω → {xs} for 0 ≤ s ≤ t

dxs = µ F(s) ds + p 2µT dBs

W(C → C0)

Diffusion

Probability of a sequence

dx0, dx1, dx2, . . . dPi = (2πdt)−1/2 exp ⇢(dBi)2 2dt

• = (4πµTdt)−1/2 exp

⇢[dxi − µF(i)]2 4µTdt

• P(ω) = lim

dt→0 ΠidPi

Diffusion + perturbation

Perturbation changes the path probability Ratio of path probabilities is finite for

dxs = µ F(s) ds+hsµ∂V ∂x (s)ds + p 2µT dBs dt → 0

P h(ω) P(ω)

Susceptibility (h>0 for s>0)

Susceptibility

hA(t)ih hA(t)i = ⌧ A(t) P h(ω) P(ω) 1

• χAV (t) = lim

h→0

hA(t)ih hA(t)i h

Generator

P h(ω) P(ω) = exp ⇢ h 2T [V (t) − V (0)] − h 2T Z t LV (s)ds

Markov generator

In this case:

L = µF ∂ ∂x + µT ∂2 ∂x2 hLV i = d dt hV i LV (x) = ⌧dV dt

• x

interpret as expected variation:

Response function

1/2 entropy production minus 1/2 “expected” entropy production

χAV (t) = Z t RAV (t, s)ds RAV (t, s) = 1 2T  d ds hV (s)A(t)i hLV (s)A(t)i

Response function

χAV (t) = Z t RAV (t, s)ds RAV (t, s) = 1 2T  d ds hV (s)A(t)i hLV (s)A(t)i

• Entropic term

Frenetic term Baiesi, Maes, Wynants, PRL (2009) Lippiello, Corberi, Zannetti, PRE (2005)

Negative response

The sum of the two terms may be <0

• Example: negative mobility for strong forces

Baerts, Basu, Maes, Safaverdi, PRE (2013)

RAV (t, s) = 1 2T  d ds hV (s)A(t)i hLV (s)A(t)i

Negative mobility

The structure of R(t,s) is different for inertial systems Achieved in a standard path-space comparison, with different drift terms (Radon-Nicodym derivative, Girsanov Th.)

dxs = µ F(s) ds + p 2µT dBs

The structure of R(t,s) is different for inertial systems Achieved in a standard path-space comparison, with different drift terms (Radon-Nicodym derivative, Girsanov Th.)

What happens if we change the noise term?

dxs = µ F(s) ds + p 2µT dBs

Mathematical problem

The response to T kicks involves changing the noise term! not well defined for different noises However: we are interested in the limit h—> 0

P h(ω) P(ω)

T(1+h), small deviation from T

dP h

t

dPt = (2πT(1 + h)dt)−1/2 exp ⇢ −[dxt − µF(t)dt]2 4µT(1 + h)dt )

• /dPt

= exp ⇢ h 2T  T + (dxt)2 2µdt + µ 2 F 2(t)dt + µT ∂F ∂x (t)dt F(t) dxt

• = exp

⇢h 2 [dS − BS dt] + h 2 [dM − BM dt]

• Dangerous term (mathematically)

entropy production, as before new terms

Four terms:

Entropy production Expected entropy production

• Activity (?)

Expected activity (?)

dS(t) = dQ(t) T = 1 T F(t) dxt BS(t)dt = − µ T  F 2(t) + T dF dx (t)

• dt

dM(xt) ≡ 1 T (dxt)2 2µdt − T

• BM(t)dt ⌘ hdMi (xt) = µ

2T F 2(t)dt

[Sekimoto, Stochastic Energetics] heat flux / temperature

Four terms:

Entropy production Expected entropy production

• Activity (?)

Expected activity (?)

dS(t) = dQ(t) T = 1 T F(t) dxt BS(t)dt = − µ T  F 2(t) + T dF dx (t)

• dt

dM(xt) ≡ 1 T (dxt)2 2µdt − T

• BM(t)dt ⌘ hdMi (xt) = µ

2T F 2(t)dt

[Sekimoto, Stochastic Energetics] heat flux / temperature

Time symmetric

Dynamical activity

Very relevant in kinetically constrained models Count the number of jumps between states time-symmetric quantity

• Lecomte, Appert-Rolland, van Wijland, PRL (2005)
• Merolle, Garrahan, D. Chandler, PNAS (2005)

Relation with activity

The (dx)^2 term should scale as the number

• f successful jumps in an underlying random

walk description

Susceptibility

χA(t) = 1 2T ⌧ A(t) h S(t) − Z t BS(s)ds + M(t) − Z t BM(s)ds i

antisymmetric (entropy produced) symmetric terms (frenetic contributions + activity) Susceptibility

Example: harmonic spring

Susceptibility of the internal energy

Inertial dynamics

Harada-Sasa/stochastic energetics for the entropy production terms

T dS(xt, vt) = mvt dvt F(xt)vtdt T BS(xt, vt) = γ m[T − mv2

t ]

dxt = vtdt m dvt = F(xt)dt − γvtdt + p 2γT dBt

Harada-Sasa for transients/jumps: Lippiello, Baiesi, Sarracino, PRL (2014)

Inertial dynamics

T dM(xt, vt) = (m dvt)2 2γdt − T − m γ F(xt)dvt T BM(xt, vt) = γ 2 " v2

t −

✓F(xt) γ ◆2#

χA(t) = 1 2T ⌧ A(t) h S(t) − Z t BS(s)ds + M(t) − Z t BM(s)ds i

antisymmetric symmetric terms Susceptibility

Conclusions

General scheme: probability of trajectories -> physics Time-symmetric quantities in response formulas Not only dissipation, but also “activation” Name(s): dynamical activity, frenesy, traffic, … Attempt to compare trajectories with different T