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Towards Dynamical Foundation of Non-Equilibrium Processes: learning from biology

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### Towards Dynamical Foundation of Non-Equilibrium Processes:learning from biology

Ping Ao

Tea-time seminar

Huzhou Normal College, Huzhou, Zhejiang

December 28, 2011

Center for Systems Biomedicine and Physics Department, Shanghai Jiao Tong University, China

(上海交通大学 系统生物医学研究院 和 物理系 敖平)

“I assign more value to discovering a fact, even about the minute thing, than to lengthy disputations on the Grand Questions that fail to lead to true understanding whatever.”

Galileo Galilei (1564-1642)

“One of the principle objects of theoretical research in any department of knowledge is to find the point of view from which the subject appears in its greatest simplicity.”

Josiah Willard Gibbs (1839–1903)

Outstanding Questions

Questions on foundation of statistical mechanics and thermodynamics (Uffink, 2005):

1) In what sense can thermodynamics said reduced to statistical mechanics?

2) How can one derive equations that are not time-reversal invariant from a time- reversal invariant dynamics?

3) How to provide a theoretical basis for the "approach to equilibrium" or irreversible processes?

Success of statistical mechanics puzzles statisticians and mathematicians (Fine, 1973):

1) Irrelevant to inference and decision-making;

2) Assured by unstated methodological practices of censoring data and selective applying arguments;

3) A result of extraordinary good fortune.

- Foundation problem already known to L. Boltzmann
- Nonlinear, no detailed balance, multiplicative noise, dissipative
- Construction with generalized Einstein relation

Word Equation of Darwin and Wallace (1858):Evolution by Variation and Selectioncandidate for (3rd) universal dynamics

Attempt for general formulation (Ao, 2005, 2008):

dx/dt = f(x) + (x, t) (trajectory perspective)

- It is evolutionary laws, not compositional laws.

Stochastic Differential Equations

Network dynamics: chemical reactions, for example

dX/dt = f(X) + (X,t) (S)

Gaussian and white noise:

< > = 0, < (X,t) (X,t’) > = 2 D (X) ε(t-t’)

X = (X1, X2, … , Xn)

Does it contain a structure which connects to fundamental symplectic structure in physics?

e.g., Does potential function exist for such dynamics?

Difficult but Important Problem encountered in many fields

- Biology:

“ … the idea that there is such a quantity (adaptive landscape—P.A.) remains one of the most widely held popular misconceptions about evolution”.

S.H. Rice, in Evolutionary Theory: mathematical and conceptual foundations (2004)

- Chemistry:

“The search for a generalized thermodynamic potential in the nonlinear range has attracted a great deal of attention, but these efforts finally failed.”

G. Nicolis in New Physics, pp332 (1989)

- Physics:

“Statistical physicists have tried to find such a variational formulation for many years because, if it existed in a useful form, it might be a powerful tool for the solution of many kinds of problems. My guess … is that no such general principle exists.”

J. Langer in Critical Problems in Physics, pp26 (1997)

and, check recent issues of Physics Today, Physical Review Letters, …

- Mathematics:

gradient vs vector systems, unsolved (Holmes, 2006)

dissipative, f 0 ; asymmetric, f 0 (absence of detailed balance);

nonlinear ; stochastic with multiplicative noise

- Economy (econophysics), finance, engineering, …

Construction, I

- Standard stochastic differential equation:

dx/dt = f(x) + (x,t) (S)

Gaussian and white, Wiener noise: < > = 0, <(x,t) (x,t’) > = 2 εD (x) (t-t’)

- The desired equation (local and trajectory view) (Ao, 2004) :

[S(x) + T(x)] dx/dt = - ψ(x) + (x,t) (N)

Gaussian and white, Wiener noise: < > = 0, <(x,t) (x,t’) > = 2 εS (x) (t-t’)

which has4 dynamical elements:

dissipative, transverse, driving, and stochastic forces.

- The desired distribution (global and ensemble view)(Yin, Ao, 2006) :

steady state (Boltzmann-Gibbs) distribution eq(x) ~ exp(- ψ(x) / ε )

Construction, II

- Describingsame phenomenon

[S(x) + T(x)] [f(x) + (x,t)] = - ψ(x) + (x,t)

- Noise and deterministic “force” have independent origins

[S(x) + T(x)] f(x) = - ψ(x)

[S(x) + T(x)] (x,t) = (x,t)

- Potential condition

[S(x) + T(x)] f(x) = 0, anti-symmetric, n(n-1)/2 equations

- Generalized Einstein relation

[S(x) + T(x)] D(x) [S(x) - T(x)] = S(x)

symmetric, n(n+1)/2 equations

- Total n2conditions for the matrix [S(x) + T(x)] !

Hence, S,T, ψ can be constructed from D and f (Inverse is easy).

Gradient Expansion

Gradient expansion (Ao, 2004; 2005)

Exact equations: ∂ × [G−1f(x)] = 0, G + Gτ= 2D.

G ≡ (S + T ) -1 = D + Q

Definition of F matrix from the “force” f in 2-d:

- F11 = ∂1f1, F12 = ∂2f1, F21 = ∂1f2, F22 = ∂2f2

Lowest order gradient expansion—linear matrix equation:

- GFτ− FGτ= 0
- G + Gτ= 2 D

Q = (FD − DFτ ) / tr(F)

- φ (x) = − ∫c dx · [G−1(x) f(x) ]

Ito, Stratonovich, and Present Methodon stochastic processes

1- d stochastic process (Ao, Kwon, Qian, Complexity, 2007)

Stochastic differential equation (pre-equation according to van Kampen):

- dt x = f (x) + ζ(x, t), Guassian-white noise, multiplicative

Ito Process (I-type):

- ∂tρ(x, t) = [ε∂x ∂x D(x) + ∂xD(x)φx(x) ] ρ(x, t)

ρ(x, t = ∞) ~ exp{ − φ (x) /ε} / D(x) ; φ (x) = - ∫ dx f(x) / D(x)

Stratanovich process (S-type):

- ∂tρ(x, t) = [ε∂xD ½ (x)∂xD ½ (x) + ∂xD(x)φx(x) ] ρ(x, t)

ρ(x, t = ∞) ~ exp{ − φ (x) /ε} / D ½ (x)

Present process (A-type):

- ∂tρ(x, t) = [ ε∂xD(x)∂x+ ∂xD(x)φx(x) ] ρ(x, t)

ρ(x, t = ∞) ~ exp{ − φ (x) /ε}

From Darwinian Dynamics to Thermodynamics

- Questions on foundation of statistical mechanics and thermodynamics (Uffink, 2005):

1) In what sense can thermodynamics said reduced to statistical mechanics?

2) How can one derive equations that are not time-reversal invariant from a time- reversal invariant dynamics?

3) How to provide a theoretical basis for the "approach to equilibrium" or irreversible processes?

- Darwinian dynamics

[S(x,t) + T(x,t)] dx/dt = - ψ(x,t) + (x,t)

Gaussian and white, Wiener noise:

< > = 0, <(x,t) (x,t’) > = 2 εS (x,t) (t-t’)

[S(x,t) + T(x,t)] D(x,t) [S(x,t) - T(x,t)] = S(x,t)

(generalized Einstein relation)

Absence of detailed balance problem solved.

- Newtonian dynamics as a “limiting” case

“Hamilton” equation:

T(x,t) dx/dt = - ψ(x,t),

when ε = 0 and S = 0 .

Thermodynamics and Statistical Mechanics

Statistical mechanics follows

steady state (Boltzmann-Gibbs) distribution,

eq(x) ~ exp(- ψ (x) /ε ) ,

partition function, phase transitions, etc.

Thermodynamics also follows:

0th law: existence of temperature, ε

1st law: “energy” conservation, potential energy, ψ

2nd law: relative entropy always decreases,

SR(t) = - ∫ (x,t) ln ((x,t)/eq(x) )

- P. Ao, Emerging of Stochastic Dynamical Equalities and Steady State Thermodynamics from Darwinian Dynamics. Communications in Theoretical Physics (2008) .

A Set of New Predictions

- Generalized Einstein relation (Ao, 2004)

[S(x) + T(x)] D(x) [S(x) - T(x)] = S(x)

S D = 1 (Einstein, 1905) T = 0, detailed balance

- Free energy equality (Jarzynski, 1997)

< exp[ - ∫c dq · f(q) ] > = exp[ – ( F(end) – F(initial) )]

<∫c dq · f(q) > ≥ F(end) – F(initial) (thermodynamic inequality)

How Would Nature Decide?

“I assign more value to discovering a fact, even about the minute thing, than to lengthy

disputations on the Grand Questions that fail to lead to true understanding whatever.”

Galileo Galilei (1564-1642)

First new experimental evidence:

Influence of Noise on Force Measurements.

Giovanni Volpe, Laurent Helden, Thomas Brettschneider, Jan

Wehr, and Clemens Bechinger.

PRL 104, 170602 (2010)

FIG. 1 (color online). (a) A Brownian particle (drawn not to scale) diffuses near a wall in

the presence of gravitational and electrostatic forces. Its trajectory perpendicular to the

wall is measured with TIRM. (b) Comparison of measured (bullets) and calculated (line)

vertical diffusion coefficient as a function of the particle-wall distance. (c) Experimentally

determined probability distribution of the local drift dz for dt = 5 ms at z = 380 nm (grey).

The dashed line is a Gaussian in excellent agreement with the experimental data.

FIG. 4 (color online). Forces obtained from a drift-velocity experiment with added noise-

induced drift [see Eq. (7) with alpha = 1 (open squares) (A-type), alpha = 0:5 (open

triangles) (S-type), and alpha = 0 (open dots) (I-type)]. The solid squares represent the

forces obtained from an equilibrium measurement (same as in Fig. 2).

Related papers

Potential in Stochastic Differential Equations: Novel Construction,

P. Ao, J. Phys. A37 L25-L30 (2004).

Structure of Stochastic Dynamics near Fixed Points,

C. Kwon, P. Ao, and D.J. Thouless, Proc. Nat’l Acad. Sci. (USA) 102 (2005) 13029-13033.

On the Existence of Potential Landscape in the Evolution of Complex Systems,

P. Ao, C. Kwon, and H. Qian, Complexity 12 (2007) 19-27.

Global view of bionetwork dynamics: adaptive landscape.

P. Ao. J. Genet. Genomics 36 (2009) 63-73

Laws in Darwinian evolutionary theory.

P. Ao, Physics of Life Reviews 2 (2005) 116-156.

Emerging of stochastic dynamical equalities and steady state thermodynamics from Darwinian dynamics.

P. Ao. Commun. Theor. Phys. 49 (2008) 1073–1090.

Acknowledgements

Co-workers:

- Xiaomei Zhu, GenMath, Seattle, USA
- Lan Yin, Beijing Univ., Beijing, P.R. China
- David Thouless, Physics, Univ. Washington
- Hong Qian, Applied Math, Univ. Washington
- Chulan Kwon, Myongji Univ., S. Korea
- Shuyun Jiao, Shanghai Jiao Tong U.
- Ruoshi Yuan, Shanghai Jiao Tong U.
- Bo Yuan, Shanghai Jiao Tong U.

Funding:

- China: 985, 973

Thank You!

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