Non-Equilibrium Thermodynamics on Networks

Matteo Polettini

Universita di Bologna

ArcidossoSeptember 8, 2009

The aim of this short talk is:

- to promote a not-well-known approach to NESM

Diffusions on graphs Schnakenberg theory of macroscopic observables

- to review its formal standard results

Linear regime Fluctuation theorem

- to inquire into new results

Perturbation theory Variational principles

- to highlight mathematical connections

Graph homology Differential geometry

Is this a CN talk?

Same background: graph theory

Different questions:

- CN: how networks behave (e.g. statistics of random graphs)

- NESM: how things behave on a networks (e.g. lattices)

Different methods:

- CN: ensembles, power laws, scaling, percolation, TL (Eq.SM)

- NESM: currents, circulation, ergodicity, LDP (stoc.proc.)

Overlapping interests:

- Graph topology and combinatorics

- Variational principles

IntroductionNon-equilibrium systems:

- dynamics: nonequilibrium dissipative/diffusive dynamics(transients, large deviations, phase transitions, first passagetimes)

- thermodynamics: characterization of steady states andmacroscopic variables

Figure: (Quasi-)steady states of macroscopic systems (convection, traffic, life)

Non-Equilibrium Stationary State (NESS):

- macroscopic dynamical forcing

- constant positive entropy production

Levels of description in NESM:

- microscopic : local transport of heat, mass, charge etc. Moreabstractly, balance of probability currents

- macroscopic : spontaneous fluctuations and response toperturbations of macroscopic observables (internal entropy,entropy production, macroscopic forces and currents)

Ref.J Schnakenberg, Network Theory of Microscopic and MacroscopicBehaviour in Master Equation Systems (1976)

Schnakenberg’s motivations come from chemistry and biology:

- chemical reaction networks: Michaelis Menten kinematics

- transport across membranes

- nerve excitations

- metabolic reaction chains

The simplest example later on. . .

Ref.J. Schnakenberg, Thermodynamic Network Analysis of BiologicalSystems (1976)

Network theory

Connected graph G � pV, Eq, V vertices and E edges (no multipleedges, no loops, but can be included. . . )

Fix an arbitrary orientation. Incidence matrix:

∇xpeq �

$&%

�1, if eÑ x

�1, if eÐ x

0, elsewhere

Contains all the topological information about the graph.

We put weights wxy on the edges (transition probabilities per unittime) and densities ρx at the vertices. All the physical informationcontained in the laplacian matrix :

∆xy �

"wxy , x � y�°

y wyx , x � y

We define the current along edge e � x Ð y :

je � wxyρy � wyxρx

Markovian evolution equation (master equation):

Btρ � ∇j � ∆ρ

Btρx �¸y

�wxyρy � wyxρx

Satisfies Bt°

k ρk � 0.

Stationary states obey Kirkhoff law

∇j� � 0 � ∆ρ�

We study topological aspects (LH) and dynamical aspects (RH).

The system is ergodic (ρt Ñ ρ�) if it is accessible andcommunicating

The dominating eigenvalue of the Laplacian matrix determines theapproach to the stationary state.

Equilibrium

All currents vanish: detailed balance

wxy

wyx�ρ0x

ρ0y

Balancing is local:

Figure: Local exchange of information at equilibrium

Figure: Path independence at equilibrium

Conservativity:

ρ0x � ρ0

x0

¹ePγ

γ:xÐx0

we

we�1

, ρx0 normalization

Equilibrium holds iff Kolmogorov criterion on circuitations:

ApCq �¸ePC

logwe

we�1

� 0, @ circuits C

Non-equilibrium

Circuitations are not null, ApCq � 0, non-locality:

Hill theorem: NESS as a sum over spanning trees

ρx 9°

Tx

±ePTx

we 0

Spanning tree: maximal subgraph which spans G without cycles.

E.g. For graph

A B

D

~~~~~~~C

the following collection of oriented trees rooted in A is obtained

oo OO

//

OO

��oo

OO oo

oo

OO oo OO

OO

oo��

�������

oo OO??�������

OO OO

��

�������

oo

oo

??�������

Number of trees grows exponentially with E .

Entropy production

Internal entropy (Gibbs-Shannon, kB � 1):

S � �¸x

ρx ln ρx

The time derivative is not necessarily positive, it has to becompleted with an heat flux:

σ �dS

dt� σenv �

¸x y

jxÐyhkkkkkkkkkikkkkkkkkkj�wyxρx � wxyρy

def. axÐyhkkkikkkjln

wyxρx

wxyρy

�¸ePE

jeae

We have so defined the microscopic force.

Macroscopic observables

Spanning tree:

maximal subgraph which spans G without cycles.

Foundamental cycles:

adding any of the E � V � 1 remaining chords eα to thetree, and isolating the cycle.

All cycles are integer combinations of the foundamental basis Cα,α � 1, . . . ,E � V � 1 (graph homology).

Algebraically speaking, cycles are vectors spanning eigenspacerelative to eigenvalue 0 of the co-laplacian matrix C � ∇T∇:

Cef �

$'''&'''%

�1,eÑ

fÐ,

eÐ

fÑ

�1,eÐ

fÐ,

eÑ

fÑ

�2, e � f0, elsewhere

Kirkhoff law Cj� � 0 implies

j� �¸

cycles α

JαCα

Thanks to current conservation, microscopic currents can beintegrated up to a number E � V � 1 of foundamental currents.

Macroscopic currents and forces:

Jα � j�eα

Aα �¸

jÐkPCα

lnwjkρk

wkjρj

� lnwe1e2we2e3 . . .wen�1en

we2e1we3e2 . . .wenen�1

Forces do not depend on the state of the system ρ.

Central result (Schnakenberg theorem):

Stationary entropy is a bilinear form of macroscopiccurrents and forces.

σ� �°α JαAα

Example: simple reaction

Simplest non-equilibrium reaction

AkA�é X

kB�è B

with A and B chemiostats, X product subsance. Concentrations:|A|,|B| constant, x variable.

. . .,, ,,x � 1jjjj

)))) x,, ,,

llllx � 1ii ii

**** . . .llll

The Law of mass action prescribes:

w pAqpx � 1|xq � kA�|A| w pAqpx � 1|xq � kA�x

w pBqpx � 1|xq � kB�|B| w pBqpx � 1|xq � kB�x

Macroscopic observables:

A � log

xA ,,

x � 1B

ii

xB ,,

x � 1A

ii

� log|A|kA�kB�

|B|kA�kB�

J �kA�kB�|A| � kA�kB�|B|

kA� � kB�

σ� � AJ

Fluctuation theorem

γ � tx1, . . . , xnu: stochastic trajectory

σtγu entropy production along a trajectory

P: probability measure over paths (well-defined)

Fluctuation theorem

Ppσtγu�σqPpσtγu��σq � exp tσ

- negative entropy trajectories are exponentially disfavoured

- at equilibrium (σ � 0), time-inversion symmetry

- holds arbitrarily far from equilibrium (controversial)

- it is a Large Deviation Principle

Micro-current (spikes when a transition occurs along e � e1 Ð e2)

jepτq �n

k�1

�δxk ,e1δxk�1,e2 � δxk�1,e1δxk ,e2

�δpτ � τkq

Consider currents along the foundamental chords eα. Then the FTreads, in terms of macroscopic variables:

Pp1t

³t0 jeαpτqdτ � Jαq

Pp1t

³t0 jeαpτqdτ � �Jαq

� exp t¸α

AαJα

Ref.D Andrieux and P Gaspard, Fluctuation theorem for currents andSchnakenberg network theory

Linear regimePerturbation of equilibrium

wxy � w 0xy � εxy

where w 0 satisfy detailed balance. Linear regime

Aα �¸β

LαβJβ

with L linear response matrix

dissipationhkkikkjLαβ �

1

2

» 8�8

dτ

fluctuationhkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkj⟨rjαpτq � xjαys rjβpτq � xjβys

⟩eq

Onsager reciprocity relations

Lαβ � Lβα

Perturbation theory?

Warning: in the following ρ�, ρ0 un-normalized.

Deletion-contraction formula for edge e : e1 Ð e2

ρ�e1pGq � ρ�e1

pGzeq � w�1e ρ�e1

pG{eq

Figure: Graph, edge deletion and edge contraction.

A modest proposal: could we use this and similar formulas to workout a perturbative expansion for the stationary state nearequilibrium?

For example, local perturbation:

"we � w 0

e � εewf�e � w 0

f

Then:ρ�e1

pGq � ρ0e1pGq � εe1

pw 0e1q2ρ0pG{eq

Very weak, do not know what happens to other states x � e1, e2.

Convolution formula, a sum over ”basins of attraction”

ρxpGq �¸H�V

yRHQx

ρxpHqρy pHq¸

e:yÑHwe

which yealds

ρ�x pGq � ρ0xpGq �

¸e

εe¸H�V

e2RHQe1,x

ρ0xpHqρ0

e2pHq �Opε2q

Still very hard to compute due to°H�V .

Contraction G{γ of a tree with respect to a path identifies allvertices of a path with a unique vertex γ:

Trees are well-behaved under contraction. Then scaling symmetry :

ρ�x pGq �¸

γ:xÐy

ρ�γpG{γq¹ePγ

we

Minimum entropy production

Minimim entropy production principle:

Out-of-equilibrium systems tend to stationary stateswhich minimise the rate at which entropy is produced,consistently with the external macroscopic constraintswhich prevent the system from reaching equilibrium

Minimum entropy production principle should be a restatement ofconservation laws.

Ref.E T Jaynes, The minimum entropy production principle:

We look for a variational principle for nonequilibrium currents. Wewant to variate σ with respect to j . Problem: too much freedom!

Zia, Schmittmann: NESS characterized by the collection ofantysimmetrc currents tjeu. The symmetric part

kxÐy � wxyρy � wyxρx

is arbitrary. Let’s keep it fixed!

We are thus considering a linear regime far from equilibrium:

δAα �¸β

LαβδJβ

A first result: Onsager relations hold.

Let us variate σ with respect to j , by keeping Aα � Aα fixedthrough Lagrange multipliers λα:

δ

δj

�σ �

¸α

λα�Aα � Aα

� �j�� 0

We obtainj�e �

¸CαQe

λα

That is to say,

J �α � λα

Minimum entropy production currents satisfy Kirkhoff equation.

We seem to have a nice variational principle, but. . .

- Experimentally, how to realize linear variations?

- Conceptually, is ”non-equilibrium linear regime” reallynon-equilibrium?

- How to test Onsager relations without variations?

As well as with FT, one has the impression of never really beingthat far from equilibrium.

Gauge invariance

Let λ1, . . . λN be non-null eigenvalues of the laplacian matrix(N ¤ V , according to multiplicities). We define

CK �N¹

K�1

pC � λk1q

such that CCK � 0. Then Kirkhoff equation Cj� � 0 implies

j� � CKh

for some gauge-potential h, determined up to a gaugetransformation

h Ñ h � Cϕ

In graph-theoretic language, a choice of gaugeis a choice of a foundamental basis of cycles.

Then one can write Schnakenberg theorem as:

σ � pa�, j�q �

2Hodge2dualityhkkkkkkkkkkkkkikkkkkkkkkkkkkjpa�,CKhq � pCKa�, hq

Now CKa� obeys Kirkhoff law since CCK � 0. Perform the samereasoning as above and obtain a ”dual theorem”

σ �¸α

Fα logHα

where

Fα � pCKa�qeα

Hα � exp¸

ePCα

he

The collection of Hα are the holonomies (Wilson loops) of thegauge potential.

Diffusion on manifoldsOn a manifold M, diffusions are described by Fokker-Planck eq.

Btρ � �Bµ pAµρ� BµνBνρq

Endow M with a metric Bµν and a connection �Aµ � �BµνAν ,with covariant derivative ∇A � B � A. Then the current isj � ∇Aρ and the holonomies (Wilson loops) are

eApCαq � exp

¾Cα

A

equilibrium � flat connection

Diffusion on manifolds as a gauge theory, with (non-compact)gauge group teϕ, ϕ P Ru.

Three cases:

- A an exact form: detailed balance, conservativity

ρpxq � ρpx0q exp

»γ:xÐx0

A

- A a neither exact nor close: what becomes of the treeexpansion? A hard question.

- A a locally exact form: remarkable result

σ� �¸α

JαAα

with Aα circulations around noncontractible loops(foundamental group) and Jα suitably defined (reminescent ofChern-Simons theory).

Ref.Da-Quan Jiang, Min Qian, Min-Ping Qian, Mathematical Theoryof Non-Equilibrium Stationary States

Conclusions

Schnakenberg network theory is a great theoretical picture.

Possible practical applications:

- Perturbative expansion of NESSs

- Coarse graining of NESSs

Possible physical results:

- Variational principles (minimum entropy production)

- Onsager relations far from equilibrium

Connections with

- Graph homology/cohomology

- Combinatorial problems in graph and knot theory

- Diffusions on manifolds

- Large deviation statistics