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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 ( )

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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 ( ) 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)

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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

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Word Equation of Darwin and Wallace (1858): Evolution by Variation and Selection candidate for (3 rd ) universal dynamics Attempt for general formulation (Ao, 2005, 2008): dx/dt = f(x) + (x, t) (trajectory perspective) It is evolutionary laws, not compositional laws.

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Stochastic Differential Equations Network dynamics: chemical reactions, for example dX/dt = f(X) + (X,t) (S) Gaussian and white noise: = 0, = 2 D (X) ε (t-t) X = (X 1, X 2, …, X n ) Does it contain a structure which connects to fundamental symplectic structure in physics? e.g., Does potential function exist for such dynamics?

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Difficult but Important Problem encountered in many fields Biology: … the idea that there is such a quantity (adaptive landscapeP.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, …

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Construction, I Standard stochastic differential equation: dx/dt = f(x) + (x,t) (S) Gaussian and white, Wiener noise: = 0, = 2 ε D (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, = 2 ε S (x) (t-t) which has 4 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 )

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Construction, II Describing same 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 n 2 conditions for the matrix [S(x) + T(x)] ! Hence, S, T, ψ can be constructed from D and f (Inverse is easy).

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Gradient Expansion Gradient expansion (Ao, 2004; 2005) Exact equations: × [G 1 f(x)] = 0, G + G τ = 2D. G (S + T ) -1 = D + Q Definition of F matrix from the force f in 2-d: F 11 = 1 f 1, F 12 = 2 f 1, F 21 = 1 f 2, F 22 = 2 f 2 Lowest order gradient expansionlinear matrix equation: GF τ FG τ = 0 G + G τ = 2 D Q = (FD DF τ ) / tr(F) φ (x) = c dx · [G 1 (x) f(x) ]

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Ito, Stratonovich, and Present Method on stochastic processes 1- d stochastic process (Ao, Kwon, Qian, Complexity, 2007) Stochastic differential equation (pre-equation according to van Kampen): d t x = f (x) + ζ(x, t), Guassian-white noise, multiplicative Ito Process (I-type): t ρ(x, t) = [ε x x D(x) + x D(x)φ x (x) ] ρ(x, t) ρ(x, t = ) ~ exp{ φ (x) /ε } / D(x) ; φ (x) = - dx f(x) / D(x) Stratanovich process (S-type): t ρ(x, t) = [ε x D ½ (x) x D ½ (x) + x D(x)φ x (x) ] ρ(x, t) ρ(x, t = ) ~ exp{ φ (x) /ε } / D ½ (x) Present process (A-type): t ρ(x, t) = [ ε x D(x) x + x D(x)φ x (x) ] ρ(x, t) ρ(x, t = ) ~ exp{ φ (x) /ε }

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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, = 2 ε S (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.

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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: 0 th law: existence of temperature, ε 1 st law: energy conservation, potential energy, ψ 2 nd law: relative entropy always decreases, S R (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).

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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[ – ( F(end) – F(initial) )] F(end) – F(initial) (thermodynamic inequality)

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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 ( ) First new experimental evidence: Influence of Noise on Force Measurements. Giovanni Volpe, Laurent Helden, Thomas Brettschneider, Jan Wehr, and Clemens Bechinger. PRL 104, (2010)

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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.

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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).

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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. Natl Acad. Sci. (USA) 102 (2005) On the Existence of Potential Landscape in the Evolution of Complex Systems, P. Ao, C. Kwon, and H. Qian, Complexity 12 (2007) Global view of bionetwork dynamics: adaptive landscape. P. Ao. J. Genet. Genomics 36 (2009) Laws in Darwinian evolutionary theory. P. Ao, Physics of Life Reviews 2 (2005) Emerging of stochastic dynamical equalities and steady state thermodynamics from Darwinian dynamics. P. Ao. Commun. Theor. Phys. 49 (2008) 1073–1090.

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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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