Abstract The von Neumann entropy of a k-body-reduced density matrix γ k quantifies the entanglement between k quantum particles and the remaining ones. In this paper, we rigorously prove general properties of this entanglement entropy as a function of k; it is concave for all 1 ⩽ k ⩽ N and non-decreasing until the midpoint k ⩽ ⌊ N / 2 ⌋ . The results hold for indistinguishable quantum particles and are independent of the statistics.
Abstract We study the interaction of agents, where each one consists of an associative memory neural network trained with the same memory patterns and possibly different reinforcement-unlearning dreaming periods. Using replica methods, we obtain the rich equilibrium phase diagram of the coupled agents. It shows phases such as the student–professor phase, where only one network benefits from the interaction while the other is unaffected; a mutualism phase, where both benefit; an indifferent phase and an insufficient phase, where neither are benefited nor impaired; a phase of amensalism where one is unchanged and the other is damaged. In addition to the paramagnetic and spin glass phases, there is also one we call the reinforced delusion phase, where agents concur without having finite overlaps with memory patterns. For zero coupling constant, the model becomes the reinforcement and removal dreaming model, which without dreaming is the Hopfield model. For finite coupling and a single memory pattern, it becomes a Mattis version of the Ashkin–Teller model.
Abstract Integer counting processes increment the integer value at transitions between states of an underlying Markov process. The generator of a counting process, which depends on a parameter conjugate to the increments, defines a complex algebraic curve through its characteristic equation, and thus a compact Riemann surface. We show that the probability of a counting process can then be written as a contour integral on that Riemann surface. Several examples are discussed in detail.
Abstract In this paper, I propose a very simple statistical ‘memory model’ of one-dimensional directed polymers, which is capable of storing and retrieving a given random quenched trajectory. The model is defined in terms of the elastic string Hamiltonian with the local attractive potential between the dynamic and the quenched random strings. The average overlap between them is calculated as a function of the temperature and the strength of the attractive potential.
Abstract We measure the relaxation time of a square lattice Ising ferromagnet that is quenched to zero-temperature from supercritical initial conditions. We reveal an anomalous and seemingly overlooked timescale associated with the relaxation to ‘frozen’ two-stripe states. While close to a power law of the form ∼ L ν , we argue this timescale actually grows as ∼ L 2 ln L , with L the linear dimension of the system. We uncover the mechanism behind this scaling form by using a synthetic initial condition that replicates the late time ordering of two-stripe states, and subsequently explain it heuristically.
Abstract We introduce the reputational voter model (RVM) to account for the time-varying abilities of individuals to influence their neighbors. To understand of the RVM, we first discuss the fitness voter model (FVM), in which each voter has a fixed and distinct fitness. In a voting event where voter i is fitter than voter j , only j changes opinion. We show that the dynamics of the FVM and the voter model are identical. We next discuss the adaptive voter model (AVM), in which the influencing voter in a voting event increases its fitness by a fixed amount. The dynamics of the AVM is non-stationary and slowly crosses over to that of FVM because of the gradual broadening of the fitness distribution of the population. Finally, we treat the RVM, in which the voter i is endowed with a reputational rank r i that ranges from 1 (highest rank) to N (lowest), where N is the population size. In a voting event in which voter i outranks j , only the opinion of j changes. Concomitantly, the rank of i increases, while that of j does not change. The rank distribution remains uniform on the integers , leading to stationary dynamics. For equal number of voters in the two voting states with these two subpopulations having the same average rand, the time to reach consensus in the mean-field limit scales as . This long consensus time arises because the average rank of the minority population is typically higher than that of the majority. Thus whenever consensus is approached, this highly ranked minority tends to drive the population away from consensus.