Stochastic theory of nonequilibrium statistical mechanics

Hao Gehaoge@pku.edu.cn

1Beijing International Center for Mathematical Research2Biodynamic Optical Imaging Center

Peking University, Chinahttp://bicmr.pku.edu.cn/~gehao/

Ge, H.: Stochastic Theory of Nonequilibrium Statistical Physics (review). Advances in Mathematics(China) 43, 161-174 (2014)

BICMR: Beijing International Center for Mathematical Research

BIOPIC: Biodynamic Optical Imaging Center

Summary of Ge group

Stochastic Biophysics(Biomath)

Stochastic theory of nonequilibrium statistical mechanics

JSP06,08PRE09,10,13,14JCP12; JSTAT15

Nonequilibrium landscape theory and rate formulas

PRL09,15JRSI11; Chaos12

Stochastic modeling of biophysical systems

Statistical machine learning of single-cell data

JPCB08,13; JPA12Phys. Rep. 12Science13;Cell14;MSB15

Which kind of physical/chemical processes can be described by stochastic processes?

• Mesoscopic scale (time and space)

• Single-molecule and single-cell (subcellular) dynamics

• Trajectory perspective

Stochastic single-molecule and single-cell kinetics

Lu, et al. Science (1998)

Single-molecule enzyme kinetics

Eldar, A. and Elowitz, M. Nature (2010)

Choi, et al. Science (2008)

Single-cell kinetics

The fundamental equation in nonequilibrium thermodynamics

Ilya Prigogine(1917-2003)Nobel Prize

in 1977

Entropy production

SdSddS ie

0 k

kki XJSdepr

Clausius inequalityTQdS

Rudolf Clausius(1822-1888)

Second law of thermodynamics

0TQdSepr

rewrite

More general

Carl Eckart(1902-1973)

P.W. Bridgman(1882-1961)Nobel Prize

in 1946

Mathematical theory of nonequilibrium steady state

Min Qian (1927-)Recipient of Hua Loo-Keng Mathematics Prize (华罗庚数学奖) in 2013

Time-independent(stationary) Markov process

The simplest three-state example at steady state: single-molecule enzyme

kinetics

Reversible Michaelis-Menten enzyme kinetics

Kinetic scheme of a simple reversible enzyme. From the perspective of a single enzyme molecule, the reaction is unimolecular and cyclic.

Ge, H., Qian, M. and Qian, H.: Phys. Rep. (2012)

Ge, H.: J. Phys. Chem. B (2008)

Two reversibleMichaelis-Menten reactions

Min, et al. Nano Lett, (2005)

PS

Simulated turnover traces of a single molecule

Ge, H.: J. Phys. Chem. B (2008)

)(t

PSt :)( SPt :)( the number of occurrences of forward and backward cycles up to time t

Steady-state cycle fluxes and nonequilibrium steady state

.][][1

][][)(lim ssss

mPmS

mPP

mSS

t

ss JJ

KP

KS

KPV

KSV

ttJ

.][][1

][)(lim ;][][1

][)(lim

mPmS

mPP

t

ss

mPmS

mSS

t

ss

KP

KS

KPV

ttJ

KP

KS

KSV

ttJ

Michaelis-Menten kinetics

Chemical potential difference: )ln(][

][ln 0321

3201

ss

ss

BB JJTk

PkkkSkkkTk

Ge, H.: JPCB (2008)Ge, H., Qian, M. and Qian,H.: Phys. Rep. (2012)

00

ssJ

Waiting cycle times

ESE EP E ES EP E-2-3 -1 0 1 2 3 PS

k2k1 k3 k1 k2 k3

k-1 k-2 k-3 k-1 k-2 k-3

The kinetic scheme for computing the waiting cycle times, which also serves for molecular motors.

.1;1;1ssssssss J

TJ

TJJ

T

Generalized Haldane equality

).|()|( TTtTPTTtTP EE

Waiting cycle time T is independent of whether the enzyme completes a forward cycle or a backward cycle, although the probability weight of these two cycles might be rather different.

Carter, N. J.; Cross, R. A. Nature (2005)

Superposed distributions！

Average time course of forward and backward steps Ge, H.: J. Phys. Chem. B (2008); J. Phys. A (2012)

Jia, et al. submitted (2015): general Markovian jumping process

Ge, et al. submitted (2015): diffusion on a circle

Nonequilibrium steady state

Ge, H., et al. Phys. Rep. (2012)

321

321BPS kkk

kkklogTk

)|()|( TTtTPTTtTP EE

Ge, H. J. Phys. Chem. B (2008); J. Phys. A (2012)

Generalized Haldane Equality

Fluctuation theorem of fluctuating chemical work

Tkn

E

E BentWP

ntWP

))(())((

1)(

TktW

BeSecond law in terms of equality

Free energy conservation0

kkkkkklogTkepr

321

321B

Entropy production: Free energy dissipationFree energy input

PS 0.mEquilibriu

Traditional Second law

ttW

Transient fluctuation theorem for cyclic chemical work

2

'1

l2

l12

'1

l2

2'

1l12

'1

ltv,ltvPltv,ltvP

ltv,ltvPltv,ltvP212

1

tvtv,

'21

21etg

tgtg

tgtg

2211221

21121

log,loglog,

,log,

Ge, H.: JPA (2012)

tvtWtvtW '2211 log ,log

1eee tWtWtWtW 2121

Master equation model describing the system

No matter starting from any initial distribution, it will finally approach its stationary distribution satisfying

01

N

jij

ssiji

ssj kpkp

j

ijijiji kpkp

dttdp )(

Consider a motor protein with N different conformations R1,R2,…,RN. kij is the first-order or pseudo-first-order rate constants for the reaction Ri→Rj.

Self-assembly or self-organization

ijeqiji

eqj kpkp

Detailed balance (equilibrium state)

NESS thermodynamic force and entropy production rate

0 ji

ssij

ssij

ness AJeprT

jissjij

ssi

ssij kpkpJ

jissj

ijssi

Bssij kp

kplogTkA NESS thermodynamic force

NESS flux

NESS entropy production rate

Energy transduction efficiency at NESS

A mechanical system coupled fully reversibly to a chemical reactions, with a constant force resisting the mechanical movement driven by the chemical gradient.

l)(MechanicaOutput Energy (Chemical)Input Energy nessdh

1outputEnergy

outputEnergy inputEnergy outputEnergy

nessdh

or

(Chemical)Output Energy l)(MechanicaInput Energy nessdh

nessp

nessd Teh all dissipated! Ge, H.: PRE (2013)

From fluctuation theorems to the decomposition of epr

at transient state

Time-dependent case

j

ijijiji tkptkp

dttdp

Free energy j

kTtEB

jeTktF /)(log)(

j

kTtE

kTtEeqi

ssi j

i

eetptp /)(

/)(

)()(Boltzmann’s law

If {kij(t)} satisfys the detailed balance condition

01

N

jij

ssiji

ssj tktptktpQuasi-stationary distribution

Mathematical equivalence of Jarzynski and Hatano-Sasa equalities

Ge, H. and Qian, M., JMP (2007); Ge, H. and Jiang, D.Q., JSP (2008);

Jarzynski equality: local equilibrium

kTF

pp

kTW ee eq

/

)0()0(

/

.)0()0(

FW eqpp

Hatano-Sasa equality: without local equilibrium

1)0()0(

/

ss

ex

pp

kTQe

.kT/Q

S

ex

pp ss

00

Same theorem for time-dependent Markov process

Are these inequalities already known in the Second Law of classic thermodynamics? Do they only hold for the whole transition process between two steady states?

The traditional Clausius inequality can be in a differential form.

Decomposition of mesoscopic thermodynamic forces

tAtA

tktptktp

TktA ijssij

jij

ijiBij log

i j

0 ji

ijijp tAtJteT Entropy production

0 ji

ssijijhk tAtJtQ )()()(Housekeeping heat

Free energy dissipation 0 ji

ijijd tAtJtf )()()(

t t t dhkp fQeT Ge, H., PRE (2009);

Ge, H. and Qian, H., PRE (2010) (2013)

tktp

tktpTktA

jissj

ijssi

Bssij log

0tep for any time t In the absence of external energy input and at steady state.

0tQhk for any time t In the absence of external energy input

0tfd for any time t At steady state

Two origins of nonequilibrium

Ge, H., PRE (2009); Ge, H. and Qian, H., PRE (2010) (2013)

0 ptot e

TQ

dtdS

The new Clausius inequality is stronger than the traditional one.

0 d

hktotex fT

QQT

QdtdS

Decomposition of entropy production rate

.QfeT,Q,f

hkdp

hkd

000

Ge, H., PRE (2009);

Ge, H. and Qian, H., PRE (2010) (2013)

TQe

dtdS tot

p

Second-order stochastic system: Time-reversibility and anomalous behavior

tX,XFdt

Xdm.

2

2

Two different definitions of entropy production rates

dxdvlogmxDFF

xDk

Ts:rPTs:Plog

TlimkEPR

ttvB

PsTts

st

TB

2

01

222

001

v,xFF

Spinney, R.E., and Ford, I.J., PRL (2012); Lee, H.K., Kwon, C., and Park, H., PRL (2013)

dxdvlog

mxDvx

xDkEPR ttvB

2

2 22

Kim, K.H. and Qian, H., PRL (2004)

correspond to time-reversibility and Maxwell-Boltzmann distribution for thermodynamic equilibrium respectively

00 21 EPR,EPR

thermodynamic equilibrium Ge, H.: PRE (2014)

When the external forceis only dependent on position

xGvxv,xF xxTk

xD

B

2

021 EPREPRthermodynamic equilibrium

xUxG

;TxT

x Thermal equilibrium

Mechanical equilibrium

Flow of kinetic energy along the spatial coordinate

t,xQt,xWJxEdtd kinetic

xxkinetict

00 qx JJEPR

xkinetict

kineticxq vxEJxJ (measurable) Heat flux

Ge, H.: PRE (2014)

Thermodynamic equivalence between mesoscopic and macroscopic scales

The entropy production rate in the small-noise limit

Celani, et al.: PRL (2012); Ge, H.: PRE (2014)

dxˆ

ˆJ

xTxEPREPR t

t

overxoverspatial

2

dxxˆxTx

xTknt

xB

over

22

62

Entropy production rate of the overdamped-limit

Anomalous contribution of EPR

Hence the overdamped approximation only keeps the dynamics rather than the second law of thermodynamics.

spatialEPREPRDecomposition of entropy production rate

Local reciprocal relation between linear coefficients

Ge, H.: JSTAT (2015)

qqqpqxq

qxqpxxoverx

XLXLxJXLXLJ

xˆxxTknL t

Bqq

32

68

xˆxxTL txx

x

xTkLL Bxqqx

2

Always hold, even for far-from-equilibrium system.

Reciprocal relation between Soret effect (thermal diffusion) and Dufour effect

Come from the second moment of velocity along each dimension.

Summary

Stochastic theory of nonequilibrium statistical mechanics is rigorously studied and further developed, including the cyclic dynamics with an unexpected equality, fluctuation theorems and decomposition of entropy production rate.

A stronger Clausius’ inequality emerges only in the nonequilibrium case;

Recently we are interested in the second-order stochastic system with temperature gradient.

Prof. Min QianPeking University

Acknowledgement

Prof. Hong QianUniversity of Washington

Fundings: NSFC, MOE of PRC

Thanks for your attention!