اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAnalytical computation of frequency distributions of path-dependent processes by means of a non-multinomial maximum entropy approach
80
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Path-dependent stochastic processes are often non-ergodic and observables can no longer be computed within the ensemble picture. The resulting mathematical difficulties pose severe limits to the analytical understanding of path-dependent processes. Their statistics is typically non-multinomial in the sense that the multiplicities of the occurrence of states is not a multinomial factor. The maximum entropy principle is tightly related to multinomial processes, non-interacting systems, and to the ensemble picture; It loses its meaning for path-dependent processes. Here we show that an equivalent to the ensemble picture exists for path-dependent processes, such that the non-multinomial statistics of the underlying dynamical process, by construction, is captured correctly in a functional that plays the role of a relative entropy. We demonstrate this for self-reinforcing Polya urn processes, which explicitly generalise multinomial statistics. We demonstrate the adequacy of this constructive approach towards non-multinomial pendants of entropy by computing frequency and rank distributions of Polya urn processes. We show how microscopic update rules of a path-dependent process allow us to explicitly construct a non-multinomial entropy functional, that, when maximized, predicts the time-dependent distribution function.
قيم البحث
اقرأ أيضاً
A rigorous derivation of nonequilibrium entropy production via the path-integral formalism is presented. Entropy production is defined as the entropy change piled in a heat reservoir as a result of a nonequilibrium thermodynamic process. It is a cent
ral quantity by which various forms of the fluctuation theorem are obtained. The two kinds of the stochastic dynamics are investigated: the Langevin dynamics for an even-parity state and the Brownian motion of a single particle. Mathematical ambiguities in deriving the functional form of the entropy production, which depends on path in state space, are clarified by using a rigorous quantum mechanical approach.
We give exact formulae for a wide family of complexity measures that capture the organization of hidden nonlinear processes. The spectral decomposition of operator-valued functions leads to closed-form expressions involving the full eigenvalue spectr
um of the mixed-state presentation of a processs epsilon-machine causal-state dynamic. Measures include correlation functions, power spectra, past-future mutual information, transient and synchronization informations, and many others. As a result, a direct and complete analysis of intrinsic computation is now available for the temporal organization of finitary hidden Markov models and nonlinear dynamical systems with generating partitions and for the spatial organization in one-dimensional systems, including spin systems, cellular automata, and complex materials via chaotic crystallography.
We study dynamical reversibility in stationary stochastic processes from an information theoretic perspective. Extending earlier work on the reversibility of Markov chains, we focus on finitary processes with arbitrarily long conditional correlations
. In particular, we examine stationary processes represented or generated by edge-emitting, finite-state hidden Markov models. Surprisingly, we find pervasive temporal asymmetries in the statistics of such stationary processes with the consequence that the computational resources necessary to generate a process in the forward and reverse temporal directions are generally not the same. In fact, an exhaustive survey indicates that most stationary processes are irreversible. We study the ensuing relations between model topology in different representations, the processs statistical properties, and its reversibility in detail. A processs temporal asymmetry is efficiently captured using two canonical unifilar representations of the generating model, the forward-time and reverse-time epsilon-machines. We analyze example irreversible processes whose epsilon-machine presentations change size under time reversal, including one which has a finite number of recurrent causal states in one direction, but an infinite number in the opposite. From the forward-time and reverse-time epsilon-machines, we are able to construct a symmetrized, but nonunifilar, generator of a process---the bidirectional machine. Using the bidirectional machine, we show how to directly calculate a processs fundamental information properties, many of which are otherwise only poorly approximated via process samples. The tools we introduce and the insights we offer provide a better understanding of the many facets of reversibility and irreversibility in stochastic processes.
We study self-organisation of collective motion as a thermodynamic phenomenon, in the context of the first law of thermodynamics. It is expected that the coherent ordered motion typically self-organises in the presence of changes in the (generalised)
internal energy and of (generalised) work done on, or extracted from, the system. We aim to explicitly quantify changes in these two quantities in a system of simulated self-propelled particles, and contrast them with changes in the systems configuration entropy. In doing so, we adapt a thermodynamic formulation of the curvatures of the internal energy and the work, with respect to two parameters that control the particles alignment. This allows us to systematically investigate the behaviour of the system by varying the two control parameters to drive the system across a kinetic phase transition. Our results identify critical regimes and show that during the phase transition, where the configuration entropy of the system decreases, the rates of change of the work and of the internal energy also decrease, while their curvatures diverge. Importantly, the reduction of entropy achieved through expenditure of work is shown to peak at criticality. We relate this both to a thermodynamic efficiency and the significance of the increased order with respect to a computational path. Additionally, this study provides an information-geometric interpretation of the curvature of the internal energy as the difference between two curvatures: the curvature of the free entropy, captured by the Fisher information, and the curvature of the configuration entropy.
Maximum entropy (maxEnt) inference of state probabilities using state-dependent constraints is popular in the study of complex systems. In stochastic dynamical systems, the effect of state space topology and path-dependent constraints on the inferred
state probabilities is unknown. To that end, we derive the transition probabilities and the stationary distribution of a maximum {it path} entropy Markov process subject to state- and path-dependent constraints. The stationary distribution reflects a competition between path multiplicity and imposed constraints and is significantly different from the Boltzmann distribution. We illustrate our results with a particle diffusing on an energy landscape. Connections with the path integral approach to diffusion are discussed.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
