Nonlinear Stochastic Dynamics of Complex Systems, II: Potential of Entropic Force in Markov Systems with Nonequilibrium Steady State, Generalized Gibbs Function and Criticality
In this paper we revisit the notion of the "minus logarithm of stationary probability" as a generalized potential in nonequilibrium systems and attempt to illustrate its central role in an axiomatic approach to stochastic nonequilibrium thermodynamics of complex systems. It is demonstrated that this quantity arises naturally through both monotonicity results of Markov processes and as the rate function when a stochastic process approaches a deterministic limit. We then undertake a more detailed mathematical analysis of the consequences of this quantity, culminating in a necessary and sufficient condition for the criticality of stochastic systems. This condition is then discussed in the context of recent results about criticality in biological systems
Entropic dynamics is a framework in which the laws of dynamics are derived as an application of entropic methods of inference. Its successes include the derivation of quantum mechanics and quantum field theory from probabilistic principles. Here we develop the entropic dynamics of a system the state of which is described by a probability distribution. Thus, the dynamics unfolds on a statistical manifold which is automatically endowed by a metric structure provided by information geometry. The curvature of the manifold has a significant influence. We focus our dynamics on the statistical manifo…
Recognition that certain forces arising from the averaging of the multiple impacts of a solute particle by the surrounding solvent particles undergoing random thermal motion can be of an entropic nature has led to the incorporation of these forces and their related entropies into theoretical protocols ranging from molecular-dynamics simulations to the modeling of quarkonium suppression in particle physics. Here we present a rigorous derivation of this Brownian entropic force by means of the classical Gibbs canonical partition function and in so doing provide a heuristic demonstration of its ki…
Brain criticality through nonadditive entropic analysis of electroencephalograms
Centro Brasileiro de Pesquisas Físicas（巴西物理研究中心）；Santa Fe Institute（圣塔菲研究所）；Complexity Science Hub Vienna（维也纳复杂性科学中心）；University of Catania（卡塔尼亚大学）；Instituto Nacional da Saúde da Crianca, da Mulher e do Adolescente Fernandes Figueira. Fundação Oswaldo Cruz（费尔南德斯·菲盖拉儿童、妇女和青少年健康国家研究所。奥斯瓦尔多·克鲁兹基金会）
We investigate the critical behavior of continuous phase transitions in the context of Ginzburg Landau models with a double well effective potential. In particular, we show that the recently proposed configurational entropy, a measure of spatial complexity of the order parameter based on its Fourier mode decomposition, can be used to identify the critical point. We compute the CE for different temperatures and show that large spatial fluctuations near the critical point lead to a sharp decrease in the CE. We further show that the CE density has a marked scaling behavior near criticality, with …
Here we present the entropic dynamics formalism for networks. That is, a framework for the dynamics of graphs meant to represent a network derived from the principle of maximum entropy and the rate of transition is obtained taking into account the natural information geometry of probability distributions. We apply this framework to the Gibbs distribution of random graphs obtained with constraints on the node connectivity. The information geometry for this graph ensemble is calculated and the dynamical process is obtained as a diffusion equation. We compare the steady state of this dynamics to …
Pareto selective forces optimize several targets at the same time, instead of single fitness functions. Systems subjected to these forces evolve towards their Pareto front, a geometrical object akin to the thermodynamical Gibbs surface and whose shape and differential geometry underlie the existence of phase transitions. In this paper we outline the connection between the Pareto front and criticality and critical phase transitions. It is shown how, under definite circumstances, Pareto selective forces drive a system towards a critical ensemble that separates the two phases of a first order pha…
We propose a new way of investigating phase transitions in the context of information theory. We use an information-entropic measure of spatial complexity known as configurational entropy (CE) to quantify both the storage and exchange of information in a lattice simulation of a Ginzburg-Landau model with a scalar order parameter coupled to a heat bath. The CE is built from the Fourier spectrum of fluctuations around the mean-field and reaches a minimum at criticality. In particular, we investigate the behavior of CE near and at criticality, exploring the relation between information and the em…
In this paper we analyze the spin-wave excitations (magnons) of an inhomogeneous spin system within the Boltzmann-Gibbs framework and then connect the results with the nonextensive approach (in the sense of Tsallis statistics). Considering an equivalence between those two frameworks, we could connect the entropic parameter q with moments of the distribution of exchange integrals of the inhomogenous system. It ratifies the idea that the entropic parameter is connected to the microscopic properties of the system.
The $q$-sum $x \oplus_q y \equiv x+y+(1-q) xy$ ($x \oplus_1 y=x+y$) and the $q$-product $x\otimes_q y \equiv [x^{1-q} +y^{1-q}-1]^{\frac{1}{1-q}}$ ($x\otimes_1 y=x y$) emerge naturally within nonextensive statistical mechanics. We show here how they lead to two-parameter (namely, $q$ and $q^\prime$) generalizations of the logarithmic and exponential functions (noted respectively $\ln_{q,q^\prime}x$ and $e_{q,q^\prime}^{x}$), as well as of the Boltzmann-Gibbs-Shannon entropy $S_{BGS}\equiv -k \sum_{i=1}^Wp_i \ln p_i$ (noted $S_{q,q^\prime}$). The remarkable properties of the $(q,q^\prime)$-gene…
The Gibbs free energy is the fundamental thermodynamic potential underlying the relative stability of different states of matter under constant-pressure conditions. However, computing this quantity from atomic-scale simulations is far from trivial. As a consequence, all too often the potential energy of the system is used as a proxy, overlooking entropic and anharmonic effects. Here we discuss a combination of different thermodynamic integration routes to obtain the absolute Gibbs free energy of a solid system starting from a harmonic reference state. This approach enables the direct compariso…
Boltzmann introduced in the 1870's a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His entropic functional for classical systems was extended by Gibbs to the entire phase space of a many-body system, and by von Neumann in order to cover quantum systems as well. Finally, it was used by Shannon within the theory of information. The simplest expression of this functional corresponds to a discrete set of $W$ microscopic possibilities, and is given by $S_{BG}= -k\sum_{i=1}^W p_i \ln p_i$ ($k$ is a po…
First, we analyze trajectories inside the Feigenbaum attractor and obtain the atypical weak sensitivity to initial conditions and loss of information associated to their dynamics. We identify the Mori singularities in its Lyapunov spectrum with the appearance of a special value for the entropic index q of the Tsallis statistics. Secondly, the dynamics of iterates at the noise-perturbed transition to chaos is shown to exhibit the characteristic elements of the glass transition, e.g. two-step relaxation, aging, subdiffusion and arrest. The properties of the bifurcation gap induced by the noise a…
A general type of nonlinear Fokker-Planck equation is derived directly from a master equation, by introducing generalized transition rates. The H-theorem is demonstrated for systems that follow those classes of nonlinear Fokker-Planck equations, in the presence of an external potential. For that, a relation involving terms of Fokker-Planck equations and general entropic forms is proposed. It is shown that, at equilibrium, this relation is equivalent to the maximum-entropy principle. Families of Fokker-Planck equations may be related to a single type of entropy, and so, the correspondence betwe…
The notion of group entropy is proposed. It enables to unify and generalize many different definitions of entropy known in the literature, as those of Boltzmann-Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals are presented, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium, when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dic…
We illuminate the many-body effects underlying the structure, formation, and dissolution of cellular adhesion domains in the presence and absence of forces. We consider mixed Glauber-Kawasaki dynamics of a two-dimensional model of nearest-neighbor interacting adhesion bonds with intrinsic binding-affinity under the action of a shared pulling or pushing force. We consider adhesion bonds that are immobile due to being anchored to the underlying cytoskeleton as well as adhesion molecules that are transiently diffusing. Highly accurate analytical results are obtained on the pair-correlation level …
The generalized stochastic Loewner evolution (SLE) driven by reversible Langevin dynamics was theoretically investigated in the context of non-equilibrium statistical mechanics. The recent study of the authors revealed that the Loewner evolution enables encoding the non-equilibrium (irreversible) processes into equilibrium (reversible) processes. In this study, by Gibbs entropy-based information-theoretic approaches, we formulated this encoding mechanism of the SLE to discuss its advantages as a mean to better describe non-equilibrium states. After deriving entropy production and flux for the …
In this contribution we show that a suitably defined nonequilibrium entropy of an N-body isolated system is not a constant of the motion in general and its variation is bounded, the bounds determined by the thermodynamic entropy, i.e., the equilibrium entropy. We define the nonequilibrium entropy as a convex functional of the set of n-particle reduced distribution functions (n=0,......., N) generalizing the Gibbs fine-grained entropy formula. Additionally, as a consequence of our microscopic analysis we find that this nonequilibrium entropy behaves as a free entropic oscillator. In the approac…
It is by now well known that the Boltzmann-Gibbs (BG) entropy $S_{BG}=-k\sum_{i=1}^W p_i \ln p_i$ can be usefully generalized into the entropy $S_q=k (1-\sum_{i=1}^Wp_i^{q}) / (q-1)$ ($q\in \mathcal{R}; S_1=S_{BG}$). Microscopic dynamics determines, given classes of initial conditions, the occupation of the accessible phase space (or of a symmetry-determined nonzero-measure part of it), which in turn appears to determine the entropic form to be used. This occupation might be a uniform one (the usual {\it equal probability hypothesis} of BG statistical mechanics), which corresponds to $q=1$; it…
Shannon and Khinchin showed that assuming four information theoretic axioms the entropy must be of Boltzmann-Gibbs type, $S=-\sum_i p_i \log p_i$. Here we note that in physical systems one of these axioms may be violated. For non-ergodic systems the so called separation axiom (Shannon-Khinchin axiom 4) will in general not be valid. We show that when this axiom is violated the entropy takes a more general form, $S_{c,d}\propto \sum_i ^W Γ(d+1, 1- c \log p_i)$, where $c$ and $d$ are scaling exponents and $Γ(a,b)$ is the incomplete gamma function. The exponents $(c,d)$ define equivalence classes …
Generalized Algebra Grounded on Nonadditive Entropies
Rio de Janeiro State University（里约热内卢州立大学）；Legislative Advisory, Federal Senate of Brazil（巴西联邦参议院立法咨询局）；Centro Brasileiro de Pesquisas Físicas（巴西物理研究中心）；National Institute of Science and Technology for Complex Systems（复杂系统科学与技术国家研究所）；Santa Fe Institute（圣塔菲研究所）；Complexity Science Hub（复杂性科学中心）
