We generalize the convex duality symmetry in Gibbs statistical ensemble formulation, between Massieus free entropy $Phi_{V,N} (beta)$ and the Gibbs entropy $varphi_{V,N}(u)$ as a function of mean internal energy $u$. The duality tells us that Gibbs t
hermodynamic entropy is to the law of large numbers (LLN) for arithmetic sample means what Shannons information entropy is to the LLN for empirical counting frequencies. Following the same logic, we identify $u$ as the conjugate variable to counting frequency, a Hamilton-Jacobi equation for Shannon entropy as an equation of state, and suggest an eigenvalue problem for modeling statistical frequencies of correlated data.
A stochastic dynamics has a natural decomposition into a drift capturing mean rate of change and a martingale increment capturing randomness. They are two statistically uncorrelated, but not necessarily independent mechanisms contributing to the over
all fluctuations of the dynamics, representing the uncertainties in the past and in the future. A generalized Einstein relation is a consequence solely because the dynamics being stationary; and the Green-Kubo formula reflects a balance between the two mechanisms. Equilibrium with reversibility is characterized by a novel covariance symmetry.
With a scalar potential and a bivector potential, the vector field associated with the drift of a diffusion is decomposed into a generalized gradient field, a field perpendicular to the gradient, and a divergence-free field. We give such decompositio
n a probabilistic interpretation by introducing cycle velocity from a bivectorial formalism of nonequilibrium thermodynamics. New understandings on the mean rates of thermodynamic quantities are presented. Deterministic dynamical system is further proven to admit a generalized gradient form with the emerged potential as the Lyapunov function by the method of random perturbations.
We generalize an idea in the works of Landauer and Bennett on computations, and Hills in chemical kinetics, to emphasize the importance of kinetic cycles in mesoscopic nonequilibrium thermodynamics (NET). For continuous stochastic systems, a NET in p
hase space is formulated in terms of cycle affinity $ ablawedgebig(mathbf{D}^{-1}mathbf{b}big)$ and vorticity potential $mathbf{A}(mathbf{x})$ of the stationary flux $mathbf{J}^{*}= ablatimesmathbf{A}$. Each bivectorial cycle couples two transport processes represented by vectors and gives rise to Onsagers reciprocality; the scalar product of the two bivectors $mathbf{A}cdot ablawedgebig(mathbf{D}^{-1}mathbf{b}big)$ is the rate of local entropy production in the nonequilibrium steady state. An Onsager operator that maps vorticity to cycle affinity is introduced.
Stochastic entropy production, which quantifies the difference between the probabilities of trajectories of a stochastic dynamics and its time reversals, has a central role in nonequilibrium thermodynamics. In the theory of probability, the change in
the statistical properties of observables can be represented by a change in the probability measure. We consider operators on the space of probability measure that induce changes in the statistical properties of a process, and formulate entropy productions in terms of these change-of-probability-measure (CPM) operators. This mathematical underpinning of the origin of entropy productions allows us to achieve an organization of various forms of fluctuation relations: All entropy productions have a non-negative mean value, admit the integral fluctuation theorem, and satisfy a rather general fluctuation relation. Other results such as the transient fluctuation theorem and detailed fluctuation theorems then are derived from the general fluctuation relation with more constraints on the operator. We use a discrete-time, discrete-state-space Markov process to draw the contradistinction among three reversals of a process: time reversal, protocol reversal and the dual process. The properties of their corresponding CPM operators are examined, and the domains of validity of various fluctuation relations for entropy productions in physics and chemistry are revealed. We also show that our CPM operator formalism can help us rather easily extend other fluctuations relations for excess work and heat, discuss the martingale properties of entropy productions, and derive the stochastic integral formulas for entropy productions in constant-noise diffusion process with Girsanov theorem. Our formalism provides a general and concise way to study the properties of entropy-related quantities in stochastic thermodynamics and information theory.
Stochastic kinematic description of a complex dynamics is shown to dictate an energetic and thermodynamic structure. An energy function $varphi(x)$ emerges as the limit of the generalized, nonequilibrium free energy of a Markovian dynamics with vanis
hing fluctuations. In terms of the $ ablavarphi$ and its orthogonal field $gamma(x)perp ablavarphi$, a general vector field $b(x)$ can be decomposed into $-D(x) ablavarphi+gamma$, where $ ablacdotbig(omega(x)gamma(x)big)=$ $- ablaomega D(x) ablavarphi$. The matrix $D(x)$ and scalar $omega(x)$, two additional characteristics to the $b(x)$ alone, represent the local geometry and density of states intrinsic to the statistical motion in the state space at $x$. $varphi(x)$ and $omega(x)$ are interpreted as the emergent energy and degeneracy of the motion, with an energy balance equation $dvarphi(x(t))/dt=gamma D^{-1}gamma-bD^{-1}b$, reflecting the geometrical $|D ablavarphi|^2+|gamma|^2=|b|^2$. The partition function employed in statistical mechanics and J. W. Gibbs method of ensemble change naturally arise; a fluctuation-dissipation theorem is established via the two leading-order asymptotics of entropy production as $epsilonto 0$. The present theory provides a mathematical basis for P. W. Andersons emergent behavior in the hierarchical structure of complexity science.
Many biological systems can sense periodical variations in a stimulus input and produce well-timed, anticipatory responses after the input is removed. Such systems show memory effects for retaining timing information in the stimulus and cannot be und
erstood from traditional synchronization consideration of passive oscillatory systems. To understand this anticipatory phenomena, we consider oscillators built from excitable systems with the addition of an adaptive dynamics. With such systems, well-timed post-stimulus responses similar to those from experiments can be obtained. Furthermore, a well-known model of working memory is shown to possess similar anticipatory dynamics when the adaptive mechanism is identified with synaptic facilitation. The last finding suggests that this type of oscillators can be common in neuronal systems with plasticity.
