Ehrenfest Urn Research Articles

On the distribution of the hitting time for the [formula omitted]–urn Ehrenfest model

In this paper, we consider the N–urn Ehrenfest model. By utilizing an auxiliary continuous-time Markov chain, we obtain the explicit formula for the Laplace transform of the hitting time from a single state to a set A of states where A satisfies some symmetric properties. After obtaining the Laplace transform, we are able to compute the high-order moments (especially, variance) for the hitting time.

Irreversibility and typicality: A simple analytical result for the Ehrenfest model

With the aid of simple analytical computations for the Ehrenfest model, we clarify some basic features of macroscopic irreversibility. The stochastic character of the model allows us to give a non-ambiguous interpretation of the general idea that irreversibility is a typical property: for the vast majority of the realizations of the stochastic process, a single trajectory of a macroscopic observable behaves irreversibly, remaining “very close” to the deterministic evolution of its ensemble average, which can be computed using probability theory. The validity of the above scenario is checked through simple numerical simulations and a rigorous proof of the typicality is provided in the thermodynamic limit.

From Power Laws to Fractional Diffusion Processes with and Without External Forces, the Non Direct Way

Abstract In this paper, a wide view on the theory of the continuous time random walk (CTRW) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW. These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.

The Stochastic Transport Dynamics of a Conserved Quantity on a Complex Network

The stochastic dynamics of conserved quantities is an emergent phenomena observed in many complex systems, ranging from social and to biological networks. Using an extension of the Ehrenfest urn model on a complex network, over which a conserved quantity is transported in a random fashion, we study the dynamics of many elementary packets transported through the network by means of a master equation approach and compare with the mean field approximation and stochastic simulations. By use of the mean field theory, it is possible to compute an approximation to the ensemble average evolution of the number of packets in each node which, in the thermodynamic limit, agrees quite well with the results of the master equation. However, the master equation gives a more complete description of the stochastic system and provides a probabilistic view of the occupation number at each node. Of particular relevance is the standard deviation of the occupation number at each node, which is not uniform for a complex network. We analyze and compare different network topologies (small world, scale free, Erdos-Renyi, among others). Given the computational complexity of directly evaluating the asymptotic, or equilibrium, occupation number probability distribution, we propose a scaling relation with the number of packets in the network, that allows to construct the asymptotic probability distributions from the network with one packet. The approximation, which relies on the same matrix found in the mean field approach, becomes increasingly more accurate for a large number of packets.

Ehrenfest Statistical Dynamics in Chemistry: Study of Decoherence Effects.

In previous works, we introduced a geometric route to define our Ehrenfest statistical dynamics (ESD) and we proved that, for a simple toy model, the resulting ESD does not preserve purity. We now take a step further: we investigate decoherence and pointer basis in the ESD model by considering some uncertainty in the degrees of freedom of a simple but realistic molecular model, consisting of two classical cores and one quantum electron. The Ehrenfest model is sometimes discarded as a valid approximation to nonadiabatic coupled quantum-classical dynamics because it does not describe the decoherence in the quantum subsystem. However, any rigorous statistical analysis of the Ehrenfest dynamics, such as the described ESD formalism, proves that decoherence exists. In this article, decoherence in ESD is studied by measuring the change in the quantum subsystem purity and by analyzing the appearance of the pointer basis to which the system decoheres, which for our example is composed of the eigenstates of the electronic Hamiltonian.

Large deviations of a long-time average in the Ehrenfest urn model

Since its inception in 1907, the Ehrenfest urn model (EUM) has served as a test bed of key concepts of statistical mechanics. Here we employ this model to study large deviations of a time-additive quantity. We consider two continuous-time versions of the EUM with K urns and N balls: with and without interactions between the balls in the same urn. We evaluate the probability distribution that the average number of balls in one urn over time T, , takes any specified value aN, where . For long observation time, , a Donsker–Varadhan large deviation principle holds: , where … denote additional parameters of the model. We calculate the rate function exactly by two different methods due to Donsker and Varadhan and compare the exact results with those obtained with a variant of WKB approximation (after Wentzel, Kramers and Brillouin). In the absence of interactions the WKB prediction for is exact for any N. In the presence of interactions the WKB method gives asymptotically exact results for . The WKB method also uncovers the (very simple) time history of the system which dominates the contribution of different time histories to .

A Spectral Analysis of Discrete-Time Quantum Walks Related to the Birth and Death Chains

In this paper, we consider a spectral analysis of discrete time quantum walks on the path. For isospectral coin cases, we show that the time averaged distribution and stationary distributions of the quantum walks are described by the pair of eigenvalues of the coins as well as the eigenvalues and eigenvectors of the corresponding random walks which are usually referred as the birth and death chains. As an example of the results, we derive the time averaged distribution of so-called Szegedy's walk which is related to the Ehrenfest model. It is represented by Krawtchouk polynomials which is the eigenvectors of the model and includes the arcsine law.

Interactions between Ehrenfest’s urns arising from group actions

The Ehrenfest diffusion model is a well-known classical physical model consisting of two urns and n balls. There is a group theoretical interpretation of the model by using the Gelfand pair $$({\mathbb {Z}}/2{\mathbb {Z}}\wr S_{n},S_{n})$$ by Diaconis and Shahshahani (Z Wahrsch Verw Gebiete 57(2):159–179, 1981). This interpretation is still valid for an r-urns generalization. Then the corresponding Gelfand pair is $$(S_{r}\wr S_{n},S_{r-1}\wr S_{n})$$ . However, in these models, there are no restrictions for ball movements, i.e., each ball can freely move to any urns. In this paper, interactions between urns arising from actions of finite groups are introduced. Degree of freedom of ball movements are restricted by finite group actions. We then show that the cutoff phenomenon occurs in some particular (yet significant and interesting) cases.

Ehrenfest urn model with interaction.

We studied the Ehrenfest urn model in which particles in the same urn interact with each other. Depending on the nature of interaction, the system undergoes a first- or second-order phase transition. The relaxation time to the equilibrium state, the Poincaré cycles of the equilibrium state and the most far-from-equilibrium state, and the duration time of the states during first-order phase transition are calculated. It was shown that the scaling behavior the Poincaré cycles could serve as an indication to the nature of phase transition, and the behavior of the ratio of duration time of the states could be strong evidence of the metastability during first-order phase transition.

Solution to urn models of pairwise interaction with application to social, physical, and biological sciences.

We investigate a family of urn models that correspond to one-dimensional random walks with quadratic transition probabilities that have highly diverse applications. Well-known instances of these two-urn models are the Ehrenfest model of molecular diffusion, the voter model of social influence, and the Moran model of population genetics. We also provide a generating function method for diagonalizing the corresponding transition matrix that is valid if and only if the underlying mean density satisfies a linear differential equation and express the eigenvector components as terms of ordinary hypergeometric functions. The nature of the models lead to a natural extension to interaction between agents in a general network topology. We analyze the dynamics on uncorrelated heterogeneous degree sequence networks and relate the convergence times to the moments of the degree sequences for various pairwise interaction mechanisms.

Computing an expected hitting time for the 3-urn Ehrenfest model via electric networks

We study a three-urn version of the Ehrenfest model. This model can be viewed as a simple random walk on the graph represented by Z3M, the M-fold direct product of the cyclic group Z3 of order 3, where M is the total number of balls distributed in the three urns. We build an electric network by placing a unit resistance on each edge of the graph. We then apply a series of circuit analysis techniques, including the series and parallel circuit laws and the Delta-Y transformation, to establish shorted triangular resistor networks. A recurrence relation is derived for the effective resistance between two corner vertices of the triangular resistor networks. The recurrence relation is then used to obtain an explicit formula for the expected hitting time between two extreme states where all balls reside in one of the three urns.

On the hypergroup property

The hypergroup property satisfied by certain reversible Markov chains can be seen as a generalization of the convolution related features of class random walks on groups. Carlen, Geronimo and Loss~\cite{MR2764893} developed a method for checking this property in the context of Jacobi eigen-polynomials. A probabilistic extension of their approach is proposed here, enabling to recover the discrete example of the biased Ehrenfest model due to Eagleson \cite{MR0328162}. Next a spectral characterization is provided for finite birth and death chains enjoying the hypergroup property with respect to one of the boundary points.

A note on a generalized Ehrenfest urn model: another look at the mean transition times

Abstract This note is motivated by Blom's work in 1989. We consider a generalized Ehrenfest urn model in which a randomly-chosen ball has a positive probability of moving from one urn to the other urn. We use recursion relations between the mean transition times to derive formulas in terms of finite sums, which are shown to be equivalent to the definite integrals obtained by Blom.

Asymptotics of Stochastic Protein Assembly Models

Protein polymerization is involved in many important biological phenomena and gives rise to spontaneous formation of amyloid fibrils or, more generally, polymers. The starting point of this process, called nucleation, exhibits an important variability among replicated experiments. To analyze the stochastic nature of this phenomenon, one of the simplest models considers two populations of chemical components: free monomers and polymerized monomers. Initially there are only monomers. There are two reactions for the polymerization of a monomer: either two monomers collide to combine into two polymerized monomers or a monomer is polymerized after the encounter of a polymerized monomer. It turns out that this simple model does not explain completely the variability observed in the experiments. This paper investigates extensions of this model to take into account other mechanisms of the polymerization process that may have an impact on fluctuations. The first variant consists in introducing a preliminary conformation step to take into account the biological fact that, before being polymerized, a monomer has two states, regular or misfolded. Only misfolded monomers can be polymerized so that the fluctuations of the number of misfolded monomers can also be a source of variability of the number of polymerized monomers. For the second variant, based on numerical considerations, the reaction rate $\alpha$ of spontaneous formation of a polymer is of the order of $N^{-\nu}$ for some large scaling variable $N$ representing the reaction volume and $\nu$ some positive constant. Asymptotic results involving different time scales are obtained for the corresponding Markov processes. First and second order results for the starting instant of nucleation are derived from these limit theorems. The proofs of the results rely on a study of a stochastic averaging principle for a model related to an Ehrenfest urn model, and also on a scaling analysis of a population model.

Memory State Transient Analysis (MSTA): A New Soft Error Rate Measurement Method for CMOS Memory Elements Based on Stochastic Analysis

We analyze the evolution of SRAM memory logic contents under irradiation by defining the memory state as the number of cells storing a given logic value (i.e. number of cells storing a logic-1). We find that the memory state evolution under irradiation follows an Ehrenfest urn model due to the constant effect of single event upsets, and that in large memories it can be associated to an Ornstein-Uhlenbeck process. Memory state transient analysis has been applied to determine the device Soft error rate for an SRAM fabricated in a 65 nm commercial CMOS process obtaining a very good correlation. Furthermore, our analysis shows that the technique is applicable to systems composed by various dissimilar memory components, providing–under certain circumstances–the specific Soft Error Rate of each component.

A Continuous-Time Ehrenfest Model with Catastrophes and Its Jump-Diffusion Approximation

We consider a continuous-time Ehrenfest model defined over the integers from -N to N, and subject to catastrophes occurring at constant rate. The effect of each catastrophe instantaneously resets the process to state 0. We investigate both the transient and steady-state probabilities of the above model. Further, the first passage time through state 0 is discussed. We perform a jump-diffusion approximation of the above model, which leads to the Ornstein-Uhlenbeck process with catastrophes. The underlying jump-diffusion process is finally studied, with special attention to the symmetric case arising when the Ehrenfest model has equal upward and downward transition rates.

Generalization of the Ehrenfest urn model to a complex network.

The Ehrenfest urn model is extended to a complex directed network, over which a conserved quantity is transported in a random fashion. The evolution of the conserved number of packets in each urn, or node of the network, is illustrated by means of a stochastic simulation. Using mean-field theory we were able to compute an approximation to the ensemble-average evolution of the number of packets in each node which, in the thermodynamic limit, agrees quite well with the results of the stochastic simulation. Using this analytic approximation we are able to find the asymptotic dynamical state of the system and the time scale to approach the equilibrium state, for different networks. The study is extended to large scale-free and small-world networks, in which the relevance of the connectivity distribution and the topology of the network for the distribution of time scales of the system is apparent. This analysis may contribute to the understanding of the transport properties in real networks subject to a perturbation, e.g., the asymptotic state and the time scale required to approach it.

Nonlinear Ehrenfest's urn model.

Ehrenfest's urn model is modified by introducing nonlinear terms in the associated transition probabilities. It is shown that these modifications lead, in the continuous limit, to a Fokker-Planck equation characterized by two competing diffusion terms, namely, the usual linear one and a nonlinear diffusion term typical of anomalous diffusion. By considering a generalized H theorem, the associated entropy is calculated, resulting in a sum of Boltzmann-Gibbs and Tsallis entropic forms. It is shown that the stationary state of the associated Fokker-Planck equation satisfies precisely the same equation obtained by extremization of the entropy. Moreover, the effects of the nonlinear contributions on the entropy production phenomenon are also analyzed.

Solution of the voter model by spectral analysis.

An exact spectral analysis of the Markov propagator for the voter model is presented for the complete graph and extended to the complete bipartite graph and uncorrelated random networks. Using a well-defined Martingale approximation in diffusion-dominated regions of phase space, which is almost everywhere for the voter model, this method is applied to compute analytically several key quantities such as exact expressions for the m time-step propagator of the voter model, all moments of consensus times, and the local times for each macrostate. This spectral method is motivated by a related method for solving the Ehrenfest urn problem and by formulating the voter model on the complete graph as an urn model. Comparisons of the analytical results from the spectral method and numerical results from Monte Carlo simulations are presented to validate the spectral method.

Asymptotic Behavior of a Randomization Test for a Response-Adaptive Design

Response-adaptive designs allow the incorporation of ethical goals in the performance of a clinical trial, and they have been thoroughly studied assuming that treatment responses follow a population model. However, in some clinical trials, population models are not appropriate and randomization tests appear as a plausible alternative to make inference. Randomization-based tests can be devised but the calculation of their exact p-values when a response-adaptive design is used to allocate patients is either time consuming or not feasible for moderate to large sample sizes and so asymptotic results become helpful. Nevertheless, these asymptotic results are not available for response-adaptive designs with good properties. The Klein allocation rule is a response-adaptive design, with good ethical and inferential properties, that generalizes the classical Ehrenfest urn design by making the replacement policy dependent on the response of the last patient. The goal of this article is to study the asymptotic distribution of a test statistic under a randomization-based approach when patients are allocated by using the Klein allocation rule.