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Register a new userUniversal Ideal Behavior and Macroscopic Work Relation of Linear Irreversible Stochastic Thermodynamics
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We revisit the Ornstein-Uhlenbeck (OU) process as the fundamental mathematical description of linear irreversible phenomena, with fluctuations, near an equilibrium. By identifying the underlying circulating dynamics in a stationary process as the natural generalization of classical conservative mechanics, a bridge between a family of OU processes with equilibrium fluctuations and thermodynamics is established through the celebrated Helmholtz theorem. The Helmholtz theorem provides an emergent macroscopic equation of state of the entire system, which exhibits a universal ideal thermodynamic behavior. Fluctuating macroscopic quantities are studied from the stochastic thermodynamic point of view and a non-equilibrium work relation is obtained in the macroscopic picture, which may facilitate experimental study and application of the equalities due to Jarzynski, Crooks, and Hatano and Sasa.
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For systems in an externally controllable time-dependent potential, the optimal protocol minimizes the mean work spent in a finite-time transition between two given equilibrium states. For overdamped dynamics which ignores inertia effects, the optimal protocol has been found to involve jumps of the control parameter at the beginning and end of the process. Including the inertia term, we show that this feature not only persists but that even delta peak-like changes of the control parameter at both boundaries make the process optimal. These results are obtained by analyzing two simple paradigmatic cases: First, a Brownian particle dragged by a harmonic optical trap through a viscous fluid and, second, a Brownian particle subject to an optical trap with time-dependent stiffness. These insights could be used to improve free energy calculations via either thermodynamic integration or fast growth methods using Jarzynskis equality.
For many real physico-chemical complex systems detailed mechanism includes both reversible and irreversible reactions. Such systems are typical in homogeneous combustion and heterogeneous catalytic oxidation. Most complex enzyme reactions include irreversible steps. The classical thermodynamics has no limit for irreversible reactions whereas the kinetic equations may have such a limit. We represent the systems with irreversible reactions as the limits of the fully reversible systems when some of the equilibrium concentrations tend to zero. The structure of the limit reaction system crucially depends on the relative rates of this tendency to zero. We study the dynamics of the limit system and describe its limit behavior as $t to infty$. If the reversible systems obey the principle of detailed balance then the limit system with some irreversible reactions must satisfy the {em extended principle of detailed balance}. It is formulated and proven in the form of two conditions: (i) the reversible part satisfies the principle of detailed balance and (ii) the convex hull of the stoichiometric vectors of the irreversible reactions does not intersect the linear span of the stoichiometric vectors of the reversible reactions. These conditions imply the existence of the global Lyapunov functionals and alow an algebraic description of the limit behavior. The thermodynamic theory of the irreversible limit of reversible reactions is illustrated by the analysis of hydrogen combustion.
We propose a method to obtain phase portraits for stochastic systems. Starting from the Fokker-Planck equation, we separate the dynamics into a convective and a diffusive part. We show that stable and unstable fixed points of the convective field correspond to maxima and minima of the stationary probability distribution if the probability current vanishes at these points. Stochastic phase portraits, which are vector plots of the convective field, therefore indicate the extrema of the stationary distribution and can be used to identify stochastic bifurcations that change the number and stability of these extrema. We show that limit cycles in stochastic phase portraits can indicate ridges of the probability distribution, and we identify a novel type of stochastic bifurcations, where the probability maximum moves to the edge of the system through a gap between the two nullclines of the convective field.
In this paper we present the concept of description of random processes in complex systems with the discrete time. It involves the description of kinetics of discrete processes by means of the chain of finite-difference non-Markov equations for time correlation functions (TCF). We have introduced the dynamic (time dependent) information Shannon entropy $S_i(t)$ where i=0,1,2,3,... as an information measure of stochastic dynamics of time correlation $(i=0)$ and time memory (i=1,2,3,...). The set of functions $S_i(t)$ constitute the quantitative measure of time correlation disorder $(i=0)$ and time memory disorder (i=1,2,3,...) in complex system. Harnessing the infinite set of orthogonal dynamic random variables on a basis of Gram-Shmidt orthogonalization procedure tends to creation of infinite chain of finite-difference non-Markov kinetic equations for discrete TCF and memory function.The solution of the equations above thereof brings to the recurrence relations between the TCF and MF of senior and junior orders. The results obtained offer considerable scope for attack on stochastic dynamics of discrete random processes in a complex systems. Application of this technique on the analysis of stochastic dynamics of RR-intervals from human ECGs shows convincing evidence for a non-Markovian phenomemena associated with a peculiarities in short and long-range scaling. This method may be of use in distinguishing healthy from pathologic data sets based in differences in these non-Markovian properties.
One of the major resource requirements of computers - ranging from biological cells to human brains to high-performance (engineered) computers - is the energy used to run them. Those costs of performing a computation have long been a focus of research in physics, going back to the early work of Landauer. One of the most prominent aspects of computers is that they are inherently nonequilibrium systems. However, the early research was done when nonequilibrium statistical physics was in its infancy, which meant the work was formulated in terms of equilibrium statistical physics. Since then there have been major breakthroughs in nonequilibrium statistical physics, which are allowing us to investigate the myriad aspects of the relationship between statistical physics and computation, extending well beyond the issue of how much work is required to erase a bit. In this paper I review some of this recent work on the `stochastic thermodynamics of computation. After reviewing the salient parts of information theory, computer science theory, and stochastic thermodynamics, I summarize what has been learned about the entropic costs of performing a broad range of computations, extending from bit erasure to loop-free circuits to logically reversible circuits to information ratchets to Turing machines. These results reveal new, challenging engineering problems for how to design computers to have minimal thermodynamic costs. They also allow us to start to combine computer science theory and stochastic thermodynamics at a foundational level, thereby expanding both.
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