Dynamics of the chemical master equation, a strip of chains of equations in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>d</mml:mi></mml:math>-dimensional space

Galstyan, Vahe; Saakian, David B.

doi:10.1103/physreve.86.011125

Cited by 12 publications

(7 citation statements)

References 31 publications

(51 reference statements)

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“…Remark 7 (1) The equation (15) has already known to the community of physics and chemistry for a long time, dated back to Gang Hu's work in early 1980s [29,18,2]. It is also known as Eikonal equation as well as Hamilton-Jacobi equation (HJE) for the stochastic chemical reaction models.…”

Section: Proofmentioning

confidence: 99%

Mathematical Formalism of Nonequilibrium Thermodynamics for Nonlinear Chemical Reaction Systems with General Rate Law

Ge¹,

Qian²

2016

J Stat Phys

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This paper studies a mathematical formalism of nonequilibrium thermodynamics for chemical reaction models with N species, M reactions, and general rate law. We establish a mathematical basis for J. W. Gibbs' macroscopic chemical thermodynamics under G. N. Lewis' kinetic law of entire equilibrium (detailed balance in nonlinear chemical kinetics). In doing so, the equilibrium thermodynamics is then naturally generalized to nonequilibrium settings without detailed balance. The kinetic models are represented by a Markovian jumping process. A generalized macroscopic chemical free energy function and its associated balance equation with nonnegative source and sink are the major discoveries. The proof is based on the large deviation principle of this type of Markov processes. A general fluctuation dissipation theorem for stochastic reaction kinetics is also proved. The mathematical theory illustrates how a novel macroscopic dynamic law can emerges from the mesoscopic kinetics in a multi-scale system.

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Section: Proofmentioning

confidence: 99%

Mathematical Formalism of Nonequilibrium Thermodynamics for Nonlinear Chemical Reaction Systems with General Rate Law

Ge¹,

Qian²

2016

J Stat Phys

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“…According to Eq. (26), the switching between the two regimes is described via a very simple law. While deriving Eq.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…24) We investigate finite populations with instructions on how the agents act are probabilistic: the whole situation changes with some probability, similar to the model of gene selfregulation. 25) In this paper we follow our recent work, 26) where we solved a modification of the master equation using the Hamilton-Jacobi equation (HJE) method. [27][28][29][30] Let us first consider a deterministic equation describing the dynamics of a large population in the context of ordinary evolutionary games.…”

Section: Introductionmentioning

confidence: 99%

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Evolutionary Games with Randomly Changing Payoff Matrices

et al. 2015

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Evolutionary games are used in various fields stretching from economics to biology. In most of these games a constant payoff matrix is assumed, although some works also consider dynamic payoff matrices. In this article we assume a possibility of switching the system between two regimes with different sets of payoff matrices. Potentially such a model can qualitatively describe the development of bacterial or cancer cells with a mutator gene present. A finite population evolutionary game is studied. The model describes the simplest version of annealed disorder in the payoff matrix and is exactly solvable at the large population limit. We analyze the dynamics of the model, and derive the equations for both the maximum and the variance of the distribution using the Hamilton-Jacobi equation formalism. The Master Equation and Its Solution via Hamilton-Jacobi EquationLet us consider the model with constant matrices: A ¼ fa ij g, B ¼ fb ij g, i; j ¼ 1; 2; . . . ; m. The total population size is N; variable X represents the number of players in the first population category. As shown in Fig. 1, the whole system can exist in two versions: the upper chain with matrix A and the lower chain with matrix B, and there are transitions between the two chains.Here we have the probability conservation condition at any moment of time τ:where PðX; Þ is the probability that the system is in state A and there are X players with the first strategy, and QðX; Þ is the probability that the system is in state B with X players with the first strategy.

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“…The complete distributions are useful to obtain the unknown parameters of the models, describing the data. We apply this method to the random walks on a strip of the chains [27], which is the case of the capital dependent Parrondo's model. We calculate the entire probability distribution for the capital, then solve the same problem for the history dependent game case.…”

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Exact probability distribution functions for Parrondo's games

2016

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We consider discrete time Brownian ratchet models: Parrondo's games. Using the Fourier transform, we calculate the exact probability distribution functions for both the capital dependent and history dependent Parrondo's games. We find that in some cases there are oscillations near the maximum of the probability distribution, and after many rounds there are two limiting distributions, for the odd and even total number of rounds of gambling. We assume that the solution of the aforementioned models can be applied to portfolio optimization.The Parrondo's games [1]- [8], related to the Brownian ratchets [9]-[15] are interesting phenomena at the intersection of game theory, econophysics and statistical physics, see [8] for the inter-disiplinary applications. In case of Brownian ratchets the particle moves in a potential, which randomly changes between 2 versions. For each there is a detailed balance condition. On average there is a motion due to the random switches between two potentials. The phenomenon is certainly related to portfolio optimization [16], [17]. In the related situation in economics, one is using the "volatility pumping" strategy in portfolio optimization, for two asset portfolios, keeping one half of the capital in the first asset, the other half in the second asset with high volatility [18].J. M. R. Parrondo invented his game following Brownian ratchets for the discrete time case [1], to model gambling. An agent uses two biased coins for the gambling, and both strategies are loosing. In some cases a random combination of two loosing games is a winning game. It is interesting that the opposite situation is also possible, a random combination of two winning games can give a loosing game.Parrondo's games have been considered either on the one dimensional axis with some periodic potential, or by looking at the time dependent version of the game parameters.For the first case the state of the system is characterized by the current value of the money X, and the choice of the strategy. There is a period M defining the rules, how capital X can move up or down. The rules of the game depend on Mod(X, M ), where M is the period of the "potential". Originally M = 3 games were considered, then M = 2 versions of Parrondo's games were constructed [7], [19]. For the history dependent versions of the game the current rules of the game depend on the past, whether there was a growth of capital in the previous rounds or not.Later many modifications of the games were invented, * Electronic address: saakian@yerphi.am i.e. different integers M for both games [20], the Allison mixture [21], where the random mixing of two random sequences creates some autocorrelation [22], and two envelope game problems [23]. Especially intriguing is the recent finding of a Parrondo's effect-like phenomenon in a Bayesian approach to the modelling of the work of a jury [24]: the unanimous decision of its members has a low confidence. All the mentioned works consider the situation with random walks, when there are random choices between differ...

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Dynamics of the chemical master equation, a strip of chains of equations ind-dimensional space

Cited by 12 publications

References 31 publications

Mathematical Formalism of Nonequilibrium Thermodynamics for Nonlinear Chemical Reaction Systems with General Rate Law

Mathematical Formalism of Nonequilibrium Thermodynamics for Nonlinear Chemical Reaction Systems with General Rate Law

Evolutionary Games with Randomly Changing Payoff Matrices

Exact probability distribution functions for Parrondo's games

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