Table of contents for issue 49, volume 52, Journal of Physics A: Mathematical and Theoretical

In this letter we show that the results of degree growth (algebraic entropy) calculations for lattice equations strongly depend on the initial value problem that one chooses. We consider two problematic types of initial value configurations, one with problems in the past light-cone, the other one causing interference in the future light-cone, and apply them to Hirota’s discrete KdV equation and to the discrete Liouville equation. Both of these initial value problems lead to exponential degree growth for Hirota’s dKdV, the quintessential integrable lattice equation. For the discrete Liouville equation, though it is linearizable, one of the initial value problems yields exponential degree growth whereas the other is shown to yield non-polynomial (though still sub-exponential) growth. These results are in contrast to the common belief that discrete integrable equations must have polynomial growth and that linearizable equations necessarily have linear degree growth, regardless of the initial value problem one imposes. Finally, as a possible remedy for one of the observed anomalies, we also propose basing integrability tests that use growth criteria on the degree growth of a single initial value instead of all the initial values.

We report results of numerical experiments which show that a family of simple one dimensional particle systems with long-range interactions, when subjected to a certain class of inelastic interaction, evolve towards states which are highly ordered in phase space, displaying particle motions which are periodic and synchronized in relative phase. For the case of a self-gravitating system, the resulting states show a remarkable stability when the inelastic interactions are turned off, with the phase space order persisting on the longest times we simulate, and much longer than the time-scale expected for relaxation of non-equilibrium states of this long-range system to thermal equilibrium. This appears to provide a novel example of ergodicity breaking in long-range systems. We propose some simple heuristic arguments which predict reasonably well properties of the synchronized states. We enlarge our study to a broader class of long-range interacting systems, where the phenomenon of synchronisation is still present, but only for systems less than a critical size. We conclude with a discussion of avenues for further investigation.

In this paper we address a model of particle diffusion, introduced by Evans and Majumdar (2011 J. Phys. A: Math. Theor. 44 435001) where, at random times, at rate , a particle resets to a position drawn from a probability distribution with density . We study an invariance law for the standard error of first-passage time observed by Reuveni (2016 Phys. Rev. Lett. 116 170601) and find that, when dealing with restarts, noise around the resetting position breaches such a law. We show that the mean first-passage time is finite and can be minimized with respect to , for a rather general class of functions . Moreover, we propose a model of a particle system evolution where the particles independently diffuse and reset but also at random times interact. We exhibit the non-equilibrium stationary distribution for the case of two interacting particles.

The exactly solvable four-vertex model with the fixed boundary conditions in the presence of an inhomogeneous linearly growing external field is considered. The partition function of the model is calculated and represented in the determinantal form. The established connection with the boxed plane partitions allows us to calculate the generating function of plane partitions with the fixed sums of their diagonals. The obtained results are another example of the connection of integrable models with enumerative combinatorics.

The Toda chain with random initial data is studied. Of particular interest are generalized Gibbs ensembles, their averaged conserved fields, and the averages of the corresponding currents. While averaged fields are well-understood, the description of averaged currents has hitherto relied on the collision-rate assumption. For the Toda chain, the rate assumption can be investigated numerically. Here, we provide convincing evidence for the validity of the rate assumption. This lends further support to the idea that generalized Euler-type equations have a structure common to all integrable extensive systems.

In Zamparo (2019 (arXiv:1903.03527)) the author has recently established sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a separable Banach space. The renewal model has been there identified with constrained and non-constrained pinning models of polymers, which amount to Gibbs changes of measure of a classical renewal process. In this paper we show that the constrained pinning model is the common mathematical structure to the Poland–Scheraga model of DNA denaturation and to some relevant one-dimensional lattice models of statistical mechanics, such as the Fisher–Felderhof model of fluids, the Wako–Saitô–Muñoz–Eaton model of protein folding, and the Tokar–Dreyssé model of strained epitaxy. Then, in the framework of the constrained pinning model, we develop an analytical characterization of the large deviation principles for cumulative rewards corresponding to multivariate deterministic rewards that are uniquely determined by, and at most of the order of magnitude of, the time elapsed between consecutive renewals. In particular, we outline the explicit calculation of the rate functions and successively we identify the conditions that prevent them from being analytic and that underlie affine stretches in their graphs. Finally, we apply the general theory to the number of renewals. From the point of view of equilibrium statistical physics and statistical mechanics, cumulative rewards of the above type are the extensive observables that enter the thermodynamic description of the system. The number of renewals, which turns out to be the commonly adopted order parameter for the Poland–Scheraga model and for also the renewal models of statistical mechanics, is one of these observables.

The standard kinetic path integral for all spatially closed Brownian paths (loops) of duration t weighted by the product mn is evaluated, where m and n are the linking numbers of the Brownian loop with two arbitrary curves in 3D space. The path integral thus indicates the extent to which these two linking numbers are correlated, ranging from the value zero for far apart curves when it is unlikely that the Brownian loop links with both, to  ±infinity for nearly coincident curves. The result takes a form that loosely resembles that for the mutual inductance of two current carrying circuits in magnetostatics, a double line integral, but is also dependent on a single extra parameter, the duration t of the path. The result for the equivalent 2D problem was given previously (Hannay 2018).

We consider solutions of the 2D Toda lattice hierarchy, which are trigonometric functions of the ‘zeroth’ time t₀  =  x. It is known that their poles move as particles of the trigonometric Ruijsenaars–Schneider model. We extend this correspondence to the level of hierarchies: the dynamics of poles with respect to the mth hierarchical time t_m (respectively, ) of the 2D Toda lattice hierarchy is shown to be governed by the Hamiltonian which is proportional to the mth Hamiltonian (respectively, ) of the Ruijsenaars–Schneider model, where L is the Lax matrix.

In this paper, we present a general framework to solve a fundamental problem in random matrix theory (RMT), i.e. the problem of describing the joint distribution of eigenvalues of the sum of two independent random Hermitian matrices and . Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of the Duistermaat–Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of n random density matrices chosen from a random state ensemble (specified in the text) increases with the number n. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components n in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.

Unitary t-designs are ‘good’ finite subsets of the unitary group that approximate the whole unitary group well. Unitary t-designs have been applied in randomized benchmarking, tomography, quantum cryptography and many other areas of quantum information science. If a unitary t-design itself is a group then it is called a unitary t-group. Although it is known that unitary t-designs in exist for any t and d, the unitary t-groups do not exist for if , as it is shown by Guralnick–Tiep (Guralnick and Tiep 2005 Represent. Theory9 138–208) and Bannai–Navarro–Rizo–Tiep (Bannai et al 2018 J. Math. Soc. Japan (accepted)). Explicit constructions of exact unitary t-designs in are not easy in general. In particular, explicit constructions of unitary 4-designs in have been an open problem in quantum information theory. We prove that some exact unitary -designs in the unitary group are constructed from unitary t-groups in that satisfy certain specific conditions. Based on this result, we specifically construct exact unitary 3-designs in from the unitary 2-group in and also unitary 4-designs in from the unitary 3-group in numerically. We also discuss some related problems and a few applications to physics problems.

We address the characterisation of non-deterministic noiseless linear amplifiers (NLAs) and compare the performances of different estimation strategies. In particular, we focus on estimation of the gain, whose value may result from a combination of different experimental effects, as those governing the overall efficiency of the detector or the transmissivity of the involved optical components. At first, we show that, contrary to naive expectations, post-selecting only the amplified states does not offer the most accurate estimate. We then focus on minimal implementations of a NLA, i.e. those obtained by coupling the input state to a two-level system, and show that the maximal amount of information about the gain of the NLA is obtained by measuring the whole composite system. The quantum Fisher information (QFI) of this best-case scenario is analysed in some detail, and compared to the QFI of the post-selected states, both for successful and unsuccessful amplification. Eventually, we show that full extraction of the available information is achieved when the non-deterministic process is implemented by a Lüders instrument. We also analyse the precision attainable by probing NLAs by single-mode pure states and measuring the field or the number of quanta, and discuss in some detail the specific cases of squeezed vacuum and coherent states.

The probability of rare and extreme events is an important quantity for design purposes. However, computing the probability of rare events can be expensive because only a few events, if any, can be observed. To this end, it is necessary to accelerate the observation of rare events using methods such as the importance splitting technique, which is the main focus here. In this work, it is shown how a genealogical importance splitting technique can be made more efficient if one knows how the rare event occurs in terms of the mean path followed by the observables. Using Monte Carlo simulations, it is shown that one can estimate this path using less rare paths. A self-similarity model is formulated and tested using an a priori and a posteriori analysis. The self-similarity principle is also tested on more complex systems including a turbulent combustion problem with 10⁷ degrees of freedom. While the self-similarity model is shown to not be strictly valid in general, it can still provide a good approximation of the rare mean paths and is a promising route for obtaining the statistics of rare events in chaotic high-dimensional systems.