Stochastic dynamics, fluctuation theorem

Several reviews have focussed on stochastic dynamics. Harris and Schiitz discuss a range of fluctuation theorems (the Jarzynski equality, the ES FR and the GC FR) in the context of stochastic, Markov systems, but in a widely applicable context. They investigate the conditions under which they apply, including an analysis of the conditions under which the GC FR is valid. In 2006, Gaspard reviewed studies where a stochastic approach to the treatment of boundaries is used to obtain FRs for the current in nanosystems, with a focus on work from his group, and also published a review on Hamiltonian systems that includes a discussion on FRs and the JE. 

Consider a system in thermal contact with a constant temperature heat bath and driven by a time-dependent process. Crooks fluctuation theorem (Crooks, 1999) is for stochastic microscopically reversible dynamics and given by... 

Crooks stationary fluctuation theorem relates entropy production to the dynamical randomness of the stochastic processes. Therefore, it relates the statistics of fluctuations to the nonequilibrium thermodynamics through the entropy production estimations. The theorem predicts that entropy production will be positive as either the system size or the observation time increases and the probability of observing an entropy production opposite to that dictated by the second law of thermodynamics decreases exponentially. 

Further large-deviation dynamical relationships are the so-called flucmation theorems, which concern the probability than some observable such as the work performed on the system would take positive or negative values under the effect of the nonequilibrium fluctuations. Since the early work of the flucmation theorem in the context of thermostated systems [52-54], stochastic [55-59] as well as Hamiltonian [60] versions have been derived. A flucmation theorem has also been derived for nonequilibrium chemical reactions [62]. A closely related result is the nonequilibrium work theorem [61] which can also be derived from the microscopic Hamiltonian dynamics. 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

In this paper we gave a dynamic extension of the DFT, by deriving a L-D equation (11) with the fluctuation-dissipation theorem (9). We showed that the stochastic equation correctly samples the density field according to the probability exp —jflf [n], (17), based on the second H-theorem (16). At this point we note however that our TO-DFT is phenomenological md it is desirable to have a first-principle dynamics generalization of DFT. 

Both deterministic and stochastic models can be defined to describe the kinetics of chemical reactions macroscopically. (Microscopic models are out of the scope of this book.) The usual deterministic model is a subclass of systems of polynomial differential equations. Qualitative dynamic behaviour of the model can be analysed knowing the structure of the reaction network. Exotic phenomena such as oscillatory, multistationary and chaotic behaviour in chemical systems have been studied very extensively in the last fifteen years. These studies certainly have modified the attitude of chemists, and exotic begins to become common . Stochastic models describe both internal and external fluctuations. In general, they are a subclass of Markovian jump processes. Two main areas are particularly emphasised, which prove the importance of stochastic aspects. First, kinetic information may be extracted from noise measurements based upon the fluctuation-dissipation theorem of chemical kinetics second, noise may change the qualitative behaviour of systems, particularly in the vicinity of instability points. 

A second catch is the noise. If one observes the movements of a colloidal particle, the Brownian motion will be evident. There may be a constant drift in the dynamics, but the movement will be irregular. Likewise, if one observes a phase-separating liquid mixture on the mesoscale, the concentration levels would not be steady, but fluctuating. The thermodynamic mean-field model neglects all fluctuations, but they can be restored in the dynamical equations, similar to added noise in particle Brownian dynamics models. The result is a set of stochastic diffusion equations, with an additional random noise source tj [20]. In principle, the value and spectrum of the noise is dictated by a fluctuation dissipation theorem, but usually one simply takes a white noise source. 

This canonical behavior of the system particles is not accounted for by standard Newtonian dynamics (where the system energy is considered to be a constant of motion). In order to perform molecular dynamics (MD) simulations of the system under the influence of thermal fluctuations, the coupling of the system to the heat bath is required. This is provided by a thermostat, i.e., by extending the equations of motion by additional heat-bath coupling degrees of freedom [75]. The introduction of thermostats into the dynamics is a notorious problem in MD and it cannot be considered to be solved satisfactorily to date [76]. In order to take into consideration the stochastic nature of any particle trajectory in the heat bath, a typical approach is to introduce random forces into the dynamics. These forces represent the collisions of system and heat-bath particles on the basis of the fluctuation-dissipation theorem. 

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