Search Results (55)

As remarked by Boltzmann, the Second Law of Thermodynamics is notable for the fact that it is readily proved using elementary statistical arguments, but becomes harder and harder to verify the more precise the microscopic description of a system. In this article, we investigate one particular realization of the 2nd Law, namely Joule heating in a wire under electrical bias. We analyze the production of entropy in an exactly solvable model of a quantum wire wherein the conserved flow of entropy under unitary quantum evolution is taken into account using an exact formula for the entropy current of a system of independent quantum particles. In this exact microscopic description of the quantum dynamics, the entropy production due to Joule heating does not arise automatically. Instead, we show that the expected entropy production is realized in the limit of a large number of local measurements by a series of floating thermoelectric probes along the length of the wire, which inject entropy into the system as a result of the information obtained via their continuous measurements of the system. The decoherence resulting from inelastic processes introduced by the local measurements is essential to the phenomenon of entropy production due to Joule heating, and would be expected to arise due to inelastic scattering in real systems of interacting particles. Full article

In this work, we investigate massive charged scalar perturbations in the background of three-dimensional dilaton black holes with a cosmological constant. We demonstrate that the wave equations governing the dynamics of these perturbations are exactly solvable, with the radial part expressible in terms of confluent Heun functions. The quasibound state frequencies are computed analytically, and we examine their dependence on the scalar field’s mass and charge, as well as on the black hole’s mass and electric charge. Our analysis also underscores the crucial role played by the cosmological constant in shaping the behavior of these perturbations. This specific black hole metric arises as a solution to the low-energy effective action of string theory in $2+1$ dimensions, and it holds potential for experimental realization in analog gravity systems due to the similarity between its surface gravity and that of acoustic analogs. Moreover, the analytic tractability of this system offers a valuable testing ground for exploring aspects of black hole spectroscopy, stability, and quantum field theory in curved spacetime. The exact solvability facilitates deeper insights into the interplay between geometry and matter fields in lower-dimensional gravity, where quantum gravitational effects can be more pronounced. Such studies not only enrich our understanding of dilaton gravity and its string-theoretic implications but also pave the way for potential applications in simulating black hole phenomena in laboratory settings using analog models. Full article

The fact that the Ising model in higher dimensions than 1D features a phase transition at the critical temperature $T_c$ despite its apparent simplicity is one of the main reasons why it has lost none of its fascination and remains a central benchmark in modeling physical systems. Building on our previous work, where an approximative analytic free-energy expression for finite 2D-Ising lattices was introduced, we investigate different extrapolation strategies for estimating $T_c$ of the infinite system from exactly solvable small lattices. Finite square lattices of linear dimension N with free and periodic boundary conditions were analyzed, exploiting their exactly accessible density of states to compute the heat capacity profiles $C\left(T\right)$. Different approaches were compared, including scaling models for the peak temperature $T_{\max }\left(N\right)$ and an envelope construction across the set of $C\left(T\right)$-profiles. We find that both approaches converge to the same asymptotic value and compare favorably to the established Binder cumulant method. Remarkably, a model for $T_{\max }$ with a single model parameter following an $N/(N+1)$-law provides robust convergence, with a physical analogy motivating this proportionality. Our findings highlight that surprisingly few, but highly accurate, finite-size results are sufficient to obtain a precise extrapolation. Full article

We investigate the thermal evolution of fermionic pairings in a finite-size SU(2) × SU(2) complex model, drawing an analogy to the BCS-BEC crossover in interacting quantum gases. Unlike the conventional crossover, which is driven by tuning the interaction strength, our study suggests that temperature alone can induce a smooth transition from weakly bound Cooper pairs (BCS-like state) to tightly bound dimers (BEC-like state). Using an exactly solvable model with a finite number of fermions, we analyze the structure of eigenstates, pairing correlations, and thermodynamic response functions. We demonstrate that different multiplet structures, characterized by distinct quasi-spin quantum numbers, become thermally accessible, effectively mimicking the crossover behavior seen in ultracold Fermi gases. Our results provide new insights into the role of thermal fluctuations in quantum pairing phenomena and suggest alternative routes for exploring crossover physics in mesoscopic and strongly correlated systems. Full article

This work introduces non-Hermitian position-dependent mass Hamiltonians characterized by complex ladder operators and real, equidistant spectra. By imposing the Heisenberg–Weyl algebraic structure as a constraint, we derive the corresponding potentials, ladder operators, and eigenfunctions. The method provides a systematic procedure for constructing exactly solvable models for arbitrary mass profiles. Specific cases are illustrated for quadratic, cosinusoidal, and exponential mass functions. Full article

In contrast to classical physics, there are not too many mathematical tools facilitating the study of singularities in quantum systems. One of the exceptions is Kato’s notion of exceptional points (EPs). Their emergence and localization are analyzed here via a family of schematic toy models. Full article

We study one-dimensional stochastic particle systems with exclusion interaction—each site can be occupied by at most one particle—and homogeneous jumping rates. Earlier work of Alimohammadi and Ahmadi classified 28 Yang–Baxter integrable two-particle interaction rules for two-species models with homogeneous rates. In this work, we show that 7 of these 28 cases can be naturally extended to integrable models with an arbitrary number of species $N\ge 2$. A key novelty of our approach is the discovery of new integrable families with one or two continuous parameters that generalize these seven cases, significantly broadening the known class of multispecies integrable exclusion processes. Furthermore, for 8 of the remaining 21 cases, we propose an alternative extension scheme that also yields integrable N-species models, thereby opening new directions for constructing and classifying integrable particle systems. Full article

We consider a multi-species mixture of interacting bosons, $N_1$ bosons of mass $m_1$, $N_2$ bosons of mass $m_2$, and $N_3$ bosons of mass $m_3$, in a harmonic trap with frequency $\omega$. The corresponding intra-species interaction strengths are ${\lambda }_{11}$, ${\lambda }_{22}$, and ${\lambda }_{33}$, and the inter-species interaction strengths are ${\lambda }_{12}$, ${\lambda }_{13}$, and ${\lambda }_{23}$. When the shape of all interactions is harmonic, the system corresponds to the generic multi-species harmonic-interaction model, which is exactly solvable. We start by solving the many-particle Hamiltonian and concisely discussing the ground-state wavefunction and energy in explicit forms as functions of all parameters, the masses, numbers of particles, and the intra-species and inter-species interaction strengths. We then explicitly compute the reduced one-particle density matrices for all the species and diagonalize them, thus generalizing the treatment by the authors earlier. The respective eigenvalues determine the degree of fragmentation of each species. As an application, we focus on phenomena that do not arise in the corresponding single-species or two-species systems. For instance, we consider a mixture of two kinds of bosons in a bath made by a third kind, controlling the fragmentation of the former by coupling to the latter. Another example exploits the possibility of different connectivities (i.e., which species interacts with which species) in the mixture, and demonstrates how the fragmentation of species 3 can be manipulated by the interaction between species 1 and species 2, when species 3 and 1 do not interact with each other. We highlight the properties of fragmentation that only appear in the multi-species mixture. Further applications are briefly discussed. Full article

In the context of the current lack of compatibility of the classical and quantum approaches to gravity, exactly solvable elementary pseudo-Hermitian quantum models are analyzed, supporting the acceptability of a point-like form of the Big Bang. The purpose is served by a hypothetical (non-covariant) identification of the “time of the Big Bang” with Kato’s exceptional-point parameter $t=0$. The consequences (including the ambiguity of the patterns of unfolding the singularity after the Big Bang) are studied in detail. In particular, singular values of the observables are shown to be useful in the analysis. Full article

This paper presents a two-qubit model derived from an SU(2)-symmetric 4×4 Hamiltonian. The resulting model is physically significant and, due to the SU(2) symmetry, is exactly solvable in both time-independent and time-dependent cases. Using the formal, general form of the related time evolution operator, the time dependence of the entanglement level for certain initial conditions is examined within the Rabi and Landau–Majorana–Stückelberg–Zener scenarios. The potential for applying this approach to higher-dimensional Hamiltonians to develop more complex exactly solvable models of interacting qubits is also highlighted. Full article

Let us consider a two-sided multi-species stochastic particle model with finitely many particles on $\mathbb{Z}$, defined as follows. Suppose that each particle is labelled by a positive integer l, and waits a random time exponentially distributed with rate 1. It then chooses the right direction to jump with probability p, or the left direction with probability $q=1-p$. If the particle chooses the right direction, it jumps to the nearest site occupied by a particle $l^{\prime }<l$ (with the convention that an empty site is considered as a particle with labelled 0). If the particle chooses the left direction, it jumps to the next site on the left only if that site is either empty or occupied by a particle $l^{\prime }<l$, and in the latter case, particles l and $l^{\prime }$ swap their positions. We show that this model is integrable, and provide the exact formula of the transition probability using the Bethe ansatz. Full article

For the first time, a self-consistent mathematical approach to describe economic processes with a general form of a memory function is proposed. In this approach, power-type memory is a special case of such general memory. The memory is described by pairs of memory functions that satisfy the Sonin and Luchko conditions. We propose using general fractional calculus (GFC) as a mathematical language that allows us to describe a general form of memory in economic processes. The existence of memory (non-locality in time) means that the process depends on the history of changes to this process in the past. Using GFC, exactly solvable economic models of natural growth with a general form of memory are proposed. Equations of natural growth with general memory are equations with general fractional derivatives and general fractional integrals for which the fundamental theorems of GFC are satisfied. Exact solutions for these equations of models of natural growth with general memory are derived. The properties of dynamic maps with a general form of memory are described in the general form and do not depend on the choice of specific types of memory functions. Examples of these solutions for various types of memory functions are suggested. Full article

A unitary-evolution process leading to an ultimate collapse and to a complete loss of observability alias quantum phase transition is studied. A specific solvable $N\mathrm{-}$state model is considered, characterized by a non-stationary non-Hermitian Hamiltonian. Our analysis uses quantum mechanics formulated in Schrödinger picture in which, in principle, only the knowledge of a complete set of observables (i.e., operators ${\mathrm{\Lambda }}_j$) enables one to guarantee the uniqueness of the related physical Hilbert space (i.e., of its inner-product metric $\mathrm{\Theta }$). Nevertheless, for the sake of simplicity, we only assume the knowledge of just a single input observable (viz., of the energy-representing Hamiltonian $H\equiv {\mathrm{\Lambda }}_1$). Then, out of all of the eligible and Hamiltonian-dependent “Hermitizing” inner-product metrics $\mathrm{\Theta }=\mathrm{\Theta }\left(H\right)$, we pick up just the simplest possible candidate. Naturally, this slightly restricts the scope of the theory, but in our present model, such a restriction is more than compensated for by the possibility of an alternative, phenomenologically better motivated constraint by which the time-dependence of the metric is required to be smooth. This opens a new model-building freedom which, in fact, enables us to force the system to reach the collapse, i.e., a genuine quantum catastrophe as a result of the mere conventional, strictly unitary evolution. Full article

Reinterpretation of mathematics behind the exactly solvable Calogero’s A-particle quantum model is used to propose its generalization. Firstly, it is argued that the strongly singular nature of Calogero’s particle–particle interactions makes the original permutation-invariant Hamiltonian tractable as a direct sum $H=⨁H_a$ of isospectral components, which are mutually independent. Secondly, after the elimination of the center-of-mass motion, the system is reconsidered as existing in the reduced Euclidean space ${\mathbb{R}}^{A-1}$ of relative coordinates and decaying into a union of subsets $W_a$ called Weyl chambers. The mutual independence of the related reduced forms of operators $H_a$ enables us to makes them nonisospectral. This breaks the symmetry and unfolds the spectral degeneracy of H. A new multiparametric generalization of the conventional A-body Calogero model is obtained. Its detailed description is provided up to $A=4$. Full article