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Deep learning machines are computational models composed of multiple processing layers of adaptive weights to learn representations of data with multiple levels of abstraction. Their structures mainly reflect the intuitive plausibility of decomposing a problem into multiple levels of computation and representation since it is believed that higher layers of representation allow a system to learn complex functions. Surprisingly, after decades of research, from learning and design perspectives these models are still deployed in a heuristic manner. In this paper, deep learning machines are modeled as disordered physical systems where its macroscopic behavior is determined in terms of the interactions defined between the basic information-processing constituent of these models, namely, the artificial neuron. They are viewed as the equilibrium states of a theoretical body that is subject to the law of increase of the entropy. The study of the changes in energy of the body when passing from one equilibrium state to another is used to understand the structure and role of the phase space of the system, and the resulting degree of disorder. It is shown that the topology of these models is strongly linked to their resulting level of disorder. Furthermore, the proposed theoretical characterization permit to assess the thermodynamic efficiency with which information can be processed by these models, and to provide a practical methodology to quantitatively estimate and compare their expected learning and generalization capabilities. These theoretical results provides new insights to the theory of deep learning and their implications are shown to be consistent through a set of benchmarks designed to experimentally assess their validity. • Methodology to model the exact partition function of any given network topology. • Criterion to quantitatively compare the performance of deep learning machines. • Topology affects learning and generalization performance and thermodynamic efficiency. • Geometrical complexity of the loss landscape linked to the potential energy content of the network. • Dynamical isometry is a necessary but not sufficient condition for learning speed. [ABSTRACT FROM AUTHOR]

In this work, we investigate thermodynamic properties and the relativistic behavior of a neutral spin-one boson particle in one-dimensional space by means of the generalized Duffin–Kemmer–Petiau (DKP) equation generated by incorporating a new nonminimal coupling related to the q -deformed formalism. Then, we obtain analytical solutions of the generalized DKP relativistic q -deformed quantum oscillator. To form the interaction of the scalar field with the q -deformed non-minimal coupling, we express the q -deformed oscillator in terms of the q -momentum and q -position operators so that these operators depend on the raising and lowering operators. In this context, we calculate some deformed thermodynamic quantities such as the partition function, Helmholtz free energy, mean energy, entropy, and specific heat capacity. [ABSTRACT FROM AUTHOR]

We investigate the Laplacian roughening model on kagome and honeycomb lattices. By using the Wang–Landau algorithm, we obtain the density of states, and calculate both the specific heat and the partition function zeros. Critical exponents and transition temperature are measured by means of the finite-size scaling analysis. The results indicate strong evidences that the system undergoes a single first-order phase transition on both lattices. On triangular and square lattices, a single second-order transition was found in our previous studies. It seems that the type of transition in the Laplacian roughening model depends on the substrate lattice structures. • The discrete Laplacian roughening model is studied on kagome and honeycomb lattices. • We calculate the specific heat and the partition function zeros. • Our results strongly support a single first-order transition. • The type of transition seems to depend on the substrate lattice structures. [ABSTRACT FROM AUTHOR]

We discuss the effects of the recent proposed Generalized Uncertainty Principle (GUP) (Perivolaropoulos, 2017), which includes a maximum length on the thermostatistics of an ideal gas. In the framework of this modified version of quantum mechanics, the deformed Schrödinger equation is solved analytically for a particle in an infinite square-well potential, then the corresponding energy spectrum is extracted. By computing the modified canonical partition function, the thermodynamic properties of the system are investigated within the canonical and microcanonical ensembles; the results show a complete consistency between both statistical descriptions. Furthermore, a comparison with the results obtained in the context of the minimal length GUP indicates that the maximum length induces fundamentally different effects, which become important at high temperatures and for large space confining the system. Especially, a modified equation of state, similar to that of real gases, emerges in the scope of this new formalism. • Quantum thermostatistics is investigated within maximum length GUP formalism. • Schrödinger's equation is solved for a particle in a box in this formalism. • Modified thermodynamic properties of an ideal gas are investigated. • Obtained results are compared with other GUP models. • New real-like equation of state is obtained and analyzed. [ABSTRACT FROM AUTHOR]

We study analytically the Ising linear chains of spin- 1 2 with the periodic conditions in nearest and next nearest neighbour interactions. The couplings for any of the nearest and next nearest neighbours are considered uniform but not necessarily equivalent to each other. The chain involves two rings of spins with different coupling constants. Therefore all or half spins contribute to interaction with next nearest neighbours depending on whether the number of spins in the chain is odd or even, respectively. We derive two exact classical solutions for the partition function of the chain in two different forms for odd and even number of spins. Besides, the pair correlations between nearest and next nearest neighbours are calculated. The influence of coupling constant corresponding to the next nearest-neighbour interactions of the ferromagnetic and antiferromagnetic chains on the thermodynamic quantities such as entropy, energy and specific heat capacity per spin at the low-temperatures is analysed. Especially, it is shown that when the nearest and next nearest neighbour interactions are of the ferromagnetic and antiferromagnetic types respectively, specific heat capacity for the chain with both odd and even number of particles exhibits the spin frustration. [ABSTRACT FROM AUTHOR]

Abstract In this investigation, the displacement operator is revisited. We established a connection between the Hermitian version of this operator with the well-known Weyl ordering. Besides, we characterized the quantum properties of a simple displaced harmonic oscillator, as well as of a displaced anisotropic two-dimensional non-Hermitian harmonic oscillator. By constructing the partition functions for both harmonic oscillators, we were able to derive several thermodynamic quantities from their energy spectra. The features of these quantities were depicted and analyzed in details. [ABSTRACT FROM AUTHOR]

The two-time correlation function for probe spin interacting with spin system (bath) is studied. We show that zeros of this function correspond to zeros of partition function of spin system in complex magnetic field. The obtained relation gives new possibility to observe the Lee–Yang zeros experimentally. Namely, we show that measuring of the time dependence of correlation function allows direct experimental observation of the Lee–Yang zeros. [ABSTRACT FROM AUTHOR]

We show that typical Renyi’s statistical mechanics’ quantifiers exhibit poles. We are referring to the partition function Z and the mean energy 〈 U 〉 . Renyi’s entropy is characterized by a real parameter α . The poles emerge in a numerable set of rational numbers belonging to the α -line. Physical effects of these poles are studied by appeal to dimensional regularization, as usual. Interesting effects are found, as for instance, gravitational ones. In particular, negative specific heats. [ABSTRACT FROM AUTHOR]

Abstract It is believed that the canonical gravitational partition function Z associated to the classical Boltzmann–Gibbs (BG) distribution e − β H Z cannot be constructed because the integral needed for building up Z includes an exponential and thus diverges at the origin. We show here that, by recourse to (1) the analytical extension treatment obtained for the first time ever, by Gradshteyn and Rizhik, via an appropriate formula for such case and (2) the dimensional regularization approach of Bollini and Giambiagi's (DR), one can indeed obtain finite gravitational results employing the BG distribution. The BG treatment is considerably more involved than its Tsallis counterpart. The latter needs only dimensional regularization, the former requires, in addition, analytical extension. Highlights • We study the q partition function of the Newtonian gravitation. • We study the poles of this partition function in phase space. • We use Dimensional Regularization of Bollini and Giambiagi. • We use an analytic extension of Gradshteyn and Rizhik. • We find interesting gravitational effects. [ABSTRACT FROM AUTHOR]

Hierarchical spin-glasses are Ising spin models defined by recursively coupling together two equally-sized sub-systems. In this work a new hierarchical spin system is introduced wherein the sub-systems are recursively coupled together through the parity of their spins. Exact Renormalization Group recursion equations for many correlators may be derived for this Hierarchical Parity Model, even for completely general couplings. Moreover, the model is computationally tractable in that O (N) algorithms exist for the computation of the partition function and ground state energy. In the special case where the couplings are all equal, the model is shown to exhibit a thermal phase transition. [ABSTRACT FROM AUTHOR]

Typical Tsallis’ statistical mechanics’ quantifiers like the partition function and the mean energy exhibit poles. We are speaking of the partition function Z and the mean energy 〈 U 〉 . The poles appear for distinctive values of Tsallis’ characteristic real parameter q , at a numerable set of rational numbers of the q -line. These poles are dealt with dimensional regularization resources. The physical effects of these poles on the specific heats are studied here for the two-body classical gravitation potential. [ABSTRACT FROM AUTHOR]

The exact analytical lattice results for the partition function of the free neutral scalar field in one spatial dimension in both the configuration and the momentum space were obtained in the framework of the path integral method. The symmetric square matrices of the bilinear forms on the vector space of fields in both configuration space and momentum space were found explicitly. The exact lattice results for the partition function were generalized to the three-dimensional spatial momentum space and the main thermodynamic quantities were derived both on the lattice and in the continuum limit. The thermodynamic properties and the finite volume corrections to the thermodynamic quantities of the free real scalar field were studied. We found that on the finite lattice the exact lattice results for the free massive neutral scalar field agree with the continuum limit only in the region of small values of temperature and volume. However, at these temperatures and volumes the continuum physical quantities for both massive and massless scalar field deviate essentially from their thermodynamic limit values and recover them only at high temperatures or/and large volumes in the thermodynamic limit. [ABSTRACT FROM AUTHOR]

To address some physical properties of percolating systems it can be useful to know the degree distributions in finite clusters along with their size distribution. Here we show that to achieve this aim for classical bond percolation one can use the q → 1 limit of suitably modified q-state Potts model. We consider a version of such model with the additional complex variables and show that its partition function gives the bond percolation’s generating function for the size and degree distribution in the q → 1 limit. For the first time we derive this distribution analytically for bond percolation on Bethe lattices and complete graph. The possibility to expand the applications of present method to other clusters’ characteristics and to models of correlated percolation is discussed. [ABSTRACT FROM AUTHOR]

The Tsallis relative entropy S q ( ρ ˆ , σ ˆ ) measures the distance between two arbitrary density matrices ρ ˆ and σ ˆ . In this work the approximation to this quantity when q = 1 + δ ( δ ≪ 1 ) is obtained. It is shown that the resulting series is equal to the von Neumann relative entropy when δ = 0 . Analyzing the von Neumann relative entropy for an arbitrary ρ ˆ and a thermal equilibrium state σ ˆ = e − β H ˆ ∕ Tr ( e − β H ˆ ) is possible to define a new inequality relating the energy, the entropy, and the partition function of the system. From this inequality, a parameter that measures the distance between the two states is defined. This distance is calculated for a general qubit system and for an arbitrary unimodal Gaussian state. In the qubit case, the dependence on the purity of the system is studied for T ≥ 0 and also for T < 0 . In the Gaussian case the general partition function, given a unimodal quadratic Hamiltonian, is calculated and the comparison of the thermal light state as a thermal equilibrium state of the parametric amplifier is presented. [ABSTRACT FROM AUTHOR]

It ts known that Tsallis’ q-non-additivity entails hidden correlations. It has also been shown that even for a monoatomic gas, both the q-partition function Z and the mean energy 〈 U 〉 diverge and, in particular, exhibit poles for certain values of the Tsallis non additivity parameter q . This happens because Z and 〈 U 〉 both depend on a Γ -function. This Γ , in turn, depends upon the spatial dimension ν . We encounter three different regimes according to the argument A of the Γ -function. (1) A > 0 , (2) A < 0 and Γ > 0 outside the poles. (3) A displays poles and the physics is obtained via dimensional regularization. In cases (2) and (3) one discovers gravitational effects and quartets of particles. Moreover, bound states and gravitational effects emerge as a consequence of the hidden q-correlations. [ABSTRACT FROM AUTHOR]

When one integrates the q -exponential function of Tsallis’ so as to get the partition function Z , a gamma function inevitably emerges. Consequently, poles arise. We investigate here the thermodynamic significance of these poles in the case of n classical harmonic oscillators (HO). Given that this is an exceedingly well known system, any new feature that may arise can safely be attributed to the poles’ effect. We appeal to the mathematical tools used in Plastino et al. (2016) and Plastino and Rocca (2017), and obtain both bound and unbound states. In the first case, we are then faced with a classical Einstein crystal. We also detect what might be interpreted as pseudo gravitational effects. [ABSTRACT FROM AUTHOR]

Knowing the exact number of particles N , and taking this knowledge into account, the quantum canonical ensemble imposes a constraint on the occupation number operators. The constraint particularly hampers the systematic calculation of the partition function and any relevant thermodynamic expectation value for arbitrary but fixed N . On the other hand, fixing only the average number of particles, one may remove the above constraint and simply factorize the traces in Fock space into traces over single-particle states. As is well known, that would be the strategy of the grand-canonical ensemble which, however, comes with an additional Lagrange multiplier to impose the average number of particles. The appearance of this multiplier can be avoided by invoking a projection operator that enables a constraint-free computation of the partition function and its derived quantities in the canonical ensemble, at the price of an angular or contour integration. Introduced in the recent past to handle various issues related to particle-number projected statistics, the projection operator approach proves beneficial to a wide variety of problems in condensed matter physics for which the canonical ensemble offers a natural and appropriate environment. In this light, we present a systematic treatment of the canonical ensemble that embeds the projection operator into the formalism of second quantization while explicitly fixing N , the very number of particles rather than the average. Being applicable to both bosonic and fermionic systems in arbitrary dimensions, transparent integral representations are provided for the partition function Z N and the Helmholtz free energy F N as well as for two- and four-point correlation functions. The chemical potential is not a Lagrange multiplier regulating the average particle number but can be extracted from F N + 1 − F N , as illustrated for a two-dimensional fermion gas. [ABSTRACT FROM AUTHOR]

‘We study the condensate fluctuation and several statistics of weakly interacting attractive Bose gas of 7 L i atoms in harmonic trap. Using exact recursion relation we calculate canonical ensemble partition function and study the thermal evolution of the condensate. As 7 L i condensate is associated with collapse, the number of condensate atom is truly finite and it facilitates to study the condensate in mesoscopic region. Being highly correlated, we utilize the two-body correlated basis function to get the many-body effective potential which is further used to calculate the energy levels. Taking van der Waals interaction as interatomic interaction we calculate several quantities like condensate fraction < n 0 > N , root-mean-square fluctuation δ n 0 and different orders of central moments. We observe the effect of finite size on the calculation of condensate fluctuations and the effect of attractive interaction over the noninteracting limit. We observe the depletion of the condensate with increase in temperature. The calculated moments nicely exhibit the mesoscopic effect. The sharp fall in the root-mean-square fluctuation near the critical point signifies the possibility of phase transition. [ABSTRACT FROM AUTHOR]

The spatial homogeneity is the basic assumption of macroscopic fundamental diagram (MFD) analysis, however, this assumption cannot be satisfied in practice. The current solutions to this problem include the network partition and using the function form of MFD that considers heterogeneity of network, while both solutions have their limitations. This paper proposes a self-adaption macroscopic fundamental diagram (SAMFD) considering network heterogeneity, which optimizes the effective road section length coefficient based on differential evolution algorithm. The effectiveness of the SAMFD is verified on the simulation data and real world datasets. The method proposed in this paper provides a new idea to solve the network heterogeneity problem when constructing MFD, and this method can be combined with the network partitioning method to better solve the heterogeneity problem. • Propose a self-adaption macroscopic fundamental diagram (SAMFD) considering network heterogeneity. • The effectiveness of the SAMFD is verified on the simulation data and real-world datasets. • SAMFD with multiple function forms has been tested. • Time stage division and road cluster partition are carried out to reduce the complexity of SAMFD. [ABSTRACT FROM AUTHOR]

Many complex systems generate multifractal time series which are long-range cross-correlated. This paper introduces three multifractal cross-correlation analysis methods, such as multifractal cross-correlation analysis based on the partition function approach (MFXPF), multifractal detrended cross-correlation analysis (MFDCCA) methods based on detrended fluctuation analysis (MFXDFA) and detrended moving average analysis (MFXDMA), which only consider one moment order. We do comparative analysis of the artificial time series (binomial multiplicative cascades and Cantor sets with different probabilities) by these methods. Then we do a feasibility test of the fixed threshold target detection within sea clutter by applying the multifractal cross-correlation analysis methods to the IPIX radar sea clutter data. The results show that it is feasible to use the method of the fixed threshold based on the multifractal feature parameter Δ f ( α ) by the MFXPF and MFXDFA-1 methods. At last, we give the main conclusions and provide a valuable reference on how to choose the multifractal algorithms, the detection parameters and the target detection methods within sea clutter in practice. [ABSTRACT FROM AUTHOR]

Complex networks as a powerful way to represent complex systems have been widely studied during the past several years. One of the most important tasks of complex network analysis is to detect communities embedded in networks. In the real world, weighted networks are very common and may contain overlapping communities where a node is allowed to belong to multiple communities. In this paper, we propose a novel Bayesian approach, called the Bayesian mixture network (BMN) model, to detect overlapping communities in weighted networks. The advantages of our method are (i) providing soft-partition solutions in weighted networks; (ii) providing soft memberships, which quantify ‘how strongly’ a node belongs to a community. Experiments on a large number of real and synthetic networks show that our model has the ability in detecting overlapping communities in weighted networks and is competitive with other state-of-the-art models at shedding light on community partition. [ABSTRACT FROM AUTHOR]

Here we apply the n -vicinity method of approximate calculation of the partition function to an Ising Model with the nearest neighbor interaction on D -dimensional hypercube lattice. We solve the equation of state for an arbitrary dimension D and analyze the behavior of the free energy. As expected, for large dimensions ( D ≥ 3 ) the system demonstrates a phase transition of the second kind. In this case, we obtain a simple analytical expression for the critical value of the inverse temperature. When 3 ≤ D ≤ 7 this expression is in a very good agreement with the results of computer simulations. In the case of small dimensions ( D = 1 , 2 ), there is a noticeable discrepancy with the known exact results. [ABSTRACT FROM AUTHOR]

We study tree approximations to classical two-body partition functions on sparse and loopy graphs via the Brydges–Kennedy–Abdessalam–Rivasseau forest expansion. We show that for sparse graphs (with large cycles), the partition function above a certain temperature T ∗ can be approximated by a graph polynomial expansion over forests of the interaction graph. Within this region, we show that the approximation can be written in terms of a reference tree T on the interaction graph, with corrections due to cycles. From this point of view, this implies that high-temperature models are easy to solve on sparse graphs, as one can evaluate the partition function using belief propagation. We also show that there exist a high- and low-temperature regime, in which T can be obtained via a maximal spanning tree algorithm on a (given) weighted graph. We study the algebra of these corrections and provide first- and second-order approximation to the tree Ansatz, and give explicit examples for the first-order approximation. [ABSTRACT FROM AUTHOR]

Topological indices (TI) are algebraic invariants of molecular graphs representing the topology of a molecule, which are very valuable in quantitative structure–property relations (QSPR). Here we prove that TI are the partition functions of such molecules when the temperature of the thermal bath at which they are submerged is very high. These partition functions are obtained by describing molecular electronic properties through tight-binding Hamiltonians (TBH), where the hopping parameters are topological properties describing atom–atom interactions. We prove that the TBH proposed here are non-Hermitian diagonalizable Hamiltonians which can be replaced by symmetric ones. In this way we propose a statistical–mechanical theory for TI, which is exemplified by deriving the Randić, Zagreb, Balaban, Wiener and ABC indices. The work also illuminates how to improve QSPR models using the current theoretical framework as well as how to derive statistical–mechanical parameters of molecular graphs. • A statistical physics interpretation of topological indices (TI) is done. • TI are the molecular partition functions at very high temperatures. • Generalized tight-binding Hamiltonians of molecular graphs are presented. • Quantitative structure–property relations can be dramatically improved. [ABSTRACT FROM AUTHOR]

We study the properties of the generalized spin-1 Ising–Heisenberg model on a diamond chain, which can be considered as a theoretical model for the homometallic magnetic complex [ Ni 3 ( C 4 H 2 O 4 ) 2 − ( μ 3 − OH ) 2 ( H 2 O ) 4 ] n ⋅ ( 2 H 2 O ) n . The model possesses a large variety of ground-state phases due to the presence of biquadratic and single-ion anisotropy parameters. Magnetization and quadrupole moment plateaus are observed at one- and two-thirds of the saturation value. The distributions of Yang–Lee and Fisher zeros are studied numerically for a variety of values of the model parameters. The usual value σ = − 1 2 alongside an unusual value σ = − 2 3 ​is determined for the Yang–Lee edge singularity exponents. [ABSTRACT FROM AUTHOR]

We study the phase behaviors of two penetrable soft spheres, whose interactions include the soft repulsion and attraction, in a hard spherical pore. The exact partition function, one-body density, and equation of state for the confined two penetrable soft spheres have been calculated using the Fourier transform method. The phase diagrams have been determined from the negative compressibility of the van der Waals type, which imitates the phase transition of many particle system. The addition of the soft repulsion and attraction beyond the soft-core potential gives rise to the van der Waals instability. The soft attraction beyond the soft-core potential significantly enhances the van der Waals instability, whereas the soft repulsion reduces the van der Waals instability. For two hard spheres and hard square-well spheres, the van der Waals instability is not observed. However, the addition of a short-range soft repulsion beyond the hard-core gives rise to the van der Waals instability. [ABSTRACT FROM AUTHOR]

The parity violation in nuclear reactions led to the discovery of the new class of toroidal multipoles. Since then, it was observed that toroidal multipoles are present in the electromagnetic structure of systems at all scales, from elementary particles, to solid state systems and metamaterials. The toroidal dipole T (the lowest order multipole) is the most common. In quantum systems, this corresponds to the toroidal dipole operator T ˆ , with the projections T ˆ i (i = 1 , 2 , 3) on the coordinate axes. Here we analyze a quantum particle in a system with cylindrical symmetry, which is a typical system in which toroidal moments appear. We find the expressions for the Hamiltonian, momenta, and toroidal dipole operators in adequate curvilinear coordinates, which allow us to find analytical expressions for the eigenfunctions of the momentum operators. While the toroidal dipole is hermitian, it is not self-adjoint, but in the new set of coordinates the operator T ˆ 3 splits into two components, one of which is (only) hermitian, whereas the other one is self-adjoint. The self-adjoint component is the one that is physically significant and represents an observable. Furthermore, we numerically diagonalize the Hamiltonian and the toroidal dipole operator and find their eigenfunctions and eigenvalues. We write the partition function and calculate the thermodynamic quantities for a system of ideal particles on a torus. Besides proving that the toroidal dipole is self-adjoint and therefore an observable, such systems open up the possibility of making metamaterials that exploit the quantization and the quantum properties of the toroidal dipoles. [ABSTRACT FROM AUTHOR]

We consider a relativistic spin 1/2 particle in the 1D-box potential in thermal non-equilibrium conditions. Using Tsallis statistics, we investigate the convexity and concavity of the deformed partition function and other thermodynamic properties of the system. We propose an upper bound to calculate the partition function integral that satisfies the Tsallis conditions. We geometrically prove that the average energy of the system depends on q -regions ( U ̃ q > 1 ≤ U ̃ q = 1 ≤ U ̃ q < 1 ). [ABSTRACT FROM AUTHOR]

The non extensive thermodynamics of an ideal gas composed by bosons and/or fermions is derived from its partition function for systems with finite chemical potentials. It is shown that the thermodynamical quantities derived in the present work are in agreement with those obtained in previous works when μ ≤ m . However some inconsistencies of previous references are corrected when μ > m . A discontinuity in the first derivatives of the partition function and its effects are discussed in detail. We show that at similar conditions, the non extensive statistics provide a harder equation of state than that provided by the Boltzmann–Gibbs statistics. [ABSTRACT FROM AUTHOR]

The partition function of q -state Potts model, or equivalently the Tutte polynomial, is computationally intractable for regular lattices. The purpose of this paper is to compute partition functions of q -state Potts model on two families of fractal lattices. Based on their self-similar structures and by applying the subgraph-decomposition method, we divide their Tutte polynomials into two summands, and for each summand we obtain a recursive formula involving the other summand. As a result, the number of spanning trees and their asymptotic growth constants, and a lower bound of the number of connected spanning subgraphs or acyclic root-connected orientations for each of such two lattices are obtained. [ABSTRACT FROM AUTHOR]

This work investigates a paths-based statistical physics formalism, inspired from the bag-of-paths framework, for the design of random walks on a graph in which the transition probabilities (the policy) are biased in favor of some node features. More precisely, given a weighted directed graph G and a non-negative cost assigned to each edge, the biased random walk is defined as the policy minimizing the expected cost rate along the walks while maintaining a constant relative entropy rate. As for the standard bag-of-paths and the randomized shortest paths frameworks, the model assigns a Gibbs–Boltzmann distribution to the set of infinite walks and allows to recover known results from the literature, derived here from a different perspective. Examples of quantities of interest are the partition function of the system, the cost rate, the optimal transition probabilities, etc. In addition, the same formalism allows the introduction of capacity constraints on the expected node visit rates and an algorithm for computing the optimal policy subject to such capacity constraints is developed. Simulation results indicate that the proposed procedure can be effectively used in order to define a Markov chain driving the walk towards nodes having some specific properties, like seniority, education level or low node degree (hub-avoiding walk). An application relying on this last property is proposed as a tool for improving serendipity in collaborative recommendation, and is tested on the MovieLens data. • This work investigates the design of an optimal biased random walk (RW) on a graph. • The biased RW is obtained by minimizing the free energy rate of the system of paths. • It optimizes a balance between relative entropy and expected cost rate of walks. • The same formalism allows to introduce capacity constraints on node visit rates. • An application to collaborative recommendation, improving serendipity, is considered. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the fractional version of the one-dimensional relativistic oscillators. We apply some important definitions and properties of a new kind of fractional formalism on the Dirac oscillator (DO). By using a semiclassical approximation, the energy eigenvalues have been determined for the oscillator. The obtained results show a remarkable influence of the fractional parameter on the energy eigenvalues. By considering a unique energy spectrum, we present a simple numerical computation of the thermal properties of a defined energy spectrum of a system. the Euler–Maclaurin formula has been used to calculate the partition function and therefore the associated thermodynamics quantities. In addition, the eigensolutions of the fractional Dirac oscillator, based on the factorization method, have been determined. [ABSTRACT FROM AUTHOR]

Abstract: We focus on the calculation of the entropy for decaying states in non-relativistic quantum mechanics. The starting point is the Friedrichs model in second quantization language. In this model, the Hamiltonian admits a spectral representation which includes resonances and Gamow states explicitly. In order to avoid the limitations posed by the definition of canonical probabilities in the presence of a complex spectrum, and/or the use of complex temperatures, we construct the partition function performing a path integration over coherent states. It is shown that the path integration yields results which are correct, at leading order, within the framework of the thermal perturbation theory. Finally, we obtain an expression for the canonical entropy of a quantum decaying system composed of fermion- and boson-states. [Copyright &y& Elsevier]

Abstract: Given an arbitrary probability distribution and a nondegenerate energy spectrum, so that a mean energy can be computed, we derive the partition function and the entropic functional that satisfy the basic relation . The procedure is illustrated by considering examples of typical distributions found currently in nature. In particular, the power-law spectrum is shown to correspond to a critical state, associated with Tsallis’ entropy. [Copyright &y& Elsevier]

Abstract: The partition function zeros of the antiferromagnetic spin- Ising–Heisenberg model on a diamond chain are studied using the transfer matrix method. Analytical equations for the distributions of Yang–Lee and Fisher zeros are derived. The Yang–Lee zeros are located on an arc of the unit circle and on the negative real axis in the complex magnetic-fugacity plane. In the limit the distribution pinches the positive real axis, precipitating a phase transition. Fisher zeros manifest more complicated distributions, depending on the values of the exchange parameters and external field. Densities of both categories of zeros are also studied. The distributions of Fisher zeros are investigated for different values of model parameters. The Yang–Lee and Fisher edge singularity exponents are shown to be identical. They are universal in nature and are calculated to be . [Copyright &y& Elsevier]

Abstract: We introduce here the -Laplace transform as a new weapon in Tsallis’ arsenal, discussing its main properties and analyzing some examples. The -Gaussian instance receives special consideration. Also, we derive the -partition function from the -Laplace transform. [Copyright &y& Elsevier]

We consider the statistical mechanical ensemble of bit string histories that are computed by a universal Turing machine. The role of the energy is played by the program size. We show that this ensemble has a first-order phase transition at a critical temperature, at which the partition function equals Chaitin's halting probability Ω. This phase transition has curious properties: the free energy is continuous near the critical temperature, but almost jumps: it converges more slowly to its finite critical value than any computable function. At the critical temperature, the average size of the bit strings diverges. We define a non-universal Turing machine that approximates this behavior of the partition function in a computable way by a super-logarithmic singularity, and discuss its thermodynamic properties. We also discuss analogies and differences between Chaitin's Omega and the partition function of a quantum mechanical particle, and with quantum Turing machines. For universal Turing machines, we conjecture that the ensemble of bit string histories at the critical temperature has a continuum formulation in terms of a string theory. • We study computations of universal Turing machines by means of statistical physics. • The "energy" of a bit string history is defined as its program size complexity. • The partition function reveals a first-order phase transition at a critical temperature. • There, it converges to Chaitin's Omega more slowly than any computable function. • At the critical point, we conjecture that bit strings are described by a string theory. [ABSTRACT FROM AUTHOR]

The method for calculating the canonical partition function with deformed Heisenberg algebra, developed by Fityo (Fityo, 2008), is adapted to the modified commutation relations including a maximal length, proposed in 1D by Perivolaropoulos (Perivolaropoulos, 2017). Firstly, the one-dimensional maximum length formalism is extended to arbitrary dimensions. Then, by employing the adapted semiclassical approach, the thermostatistics of an ideal gas and a system of harmonic oscillators (HOs) is investigated. For the ideal gas, the results generalize those obtained recently by us in 1D (Bensalem and Bouaziz, 2019), and show a complete agreement between the semiclassical and quantum approaches. In particular, a stiffer real-like equation of state for the ideal gas is established in 3D; it is consistent with the formal one, which we presented in the aforementioned paper. The modified thermostatistics of a system of HOs compared to that of an ideal gas reveals that the effects of the maximal length depend on the studied system. On the other hand, it is observed that the maximal-length effects on some thermodynamic functions of the HOs are analogous to those of the minimal length, studied previously in the literature. Finally, by analyzing some experimental data, we argue that the maximal length might be viewed as a characteristic scale associated with the system under study. • Thermostatistics with a maximal length for an ideal gas and HOs is investigated. • The one-dimensional maximum length formalism is extended to arbitrary dimensions. • Semiclassical partition functions are computed analytically for both systems. • New effects of the maximal length, which depend on the studied system, are observed. • The maximal length plays the role of a characteristic scale of the concerned system. [ABSTRACT FROM AUTHOR]

Abstract: Following the thermodynamic formulation of a multifractal measure that was shown to enable the detection of large fluctuations at an early stage, here we propose a new index which permits us to distinguish events like financial crises in real time. We calculate the partition function from which we can obtain thermodynamic quantities analogous to the free energy and specific heat. The index is defined as the normalized energy variation and it can be used to study the behavior of stochastic time series, such as financial market daily data. Famous financial market crashes–Black Thursday (1929), Black Monday (1987) and the subprime crisis (2008)–are identified with clear and robust results. The method is also applied to the market fluctuations of 2011. From these results it appears as if the apparent crisis of 2011 is of a different nature to the other three. We also show that the analysis has forecasting capabilities. [Copyright &y& Elsevier]

In this article, we study minimal length effects on the statistical thermodynamic properties of a magnetized ideal gas. We reformulate the Lagrangian density of the 3-D spatial dimensional static electromagnetic field by employing Kempf Algebra to get the modified magnetostatic potential energy. We derive the modified thermodynamic properties of a magnetized ideal gas such as the mean energy, Helmholtz free energy and entropy. Also, we derive the magnetization, susceptibility, and permeability in the presence of a minimal length. It is found that quantum gravity affects all thermodynamic properties. In addition, we derive the respective upper bounds on the deformation parameter and minimal length. • Modified thermodynamics can be useful to investigate early-universe phenomena. • The modified partition function is written in terms of the modified magnetic energy. • Modified properties are derived using the modified partition function. • All modified quantities depend on α , and the usual quantities are recovered when α → 0. [ABSTRACT FROM AUTHOR]

The thermodynamic geometry of the Bose–Einstein fluid in the framework of information geometry is revisited, and particularly the strongly degenerate case is considered for a finite volume. Therefore, in the construction of the metric, the term related to the number of particles that accumulate in the ground state is taken into account, and we allow to explore its contribution to the curvature in highly quantum conditions, namely for temperature values where the ratio λ 3 ∕ V (thermal de Broglie wavelength cubed over volume) is greater than unity. It is found that in this regime, the ground state contribution is relevant in the limit of condensation and it strongly affects the behavior of the scalar curvature R. We show, numerically and analytically, that R is finite and smoothly approaches to zero in the limit η → 1 , namely as the fugacity tends to the numerical value where the condensation occurs the quantum effects are stronger. Consequently, when both phases are taken into account, there exist a region for extremely low temperatures where the curvature is regular, similarly to other quantum phase transitions reported in the literature. • Bose–Einstein condensation in the information geometric framework is revisited. • Relevant terms in the partition function that account for particles in the ground state are considered. • The scalar curvature in standard and Gentile's statistics is calculated. • The scalar curvature results to be regular in a highly quantum regime of very low temperatures or very high densities. [ABSTRACT FROM AUTHOR]

In this comment, we discuss the mathematical formalism used in Boumali et al. (2020) which describes the superstatistical thermal properties of a one-dimensional Dirac oscillator. In particular, we point out the importance of maintaining the Legendre structure unaltered to ensure an accurate description of the thermodynamic observables when a Tsallis-like statistical description is assumed. Also, we remark that all the negative poles have to take into account to calculate the Gibbs–Boltzmann partition function. Our findings show that the divergences obtained by the authors in the Helmholtz free energy, which are propagated to the other thermal properties, are a consequence of an incomplete partition function. Moreover, we prove that the restrictions over the q -parameter are no needed if an appropriate partition function describes the system. • The super statistical framework is revisited in terms of the Legendre structure conservation. • The expansion on powers of q is corrected. • The partition function is computed by taking into account all the poles of the Mellin transform. • The thermodynamic functions have not discontinuities. • We demonstrate that there is not restrictions over q. [ABSTRACT FROM AUTHOR]

We consider the thermodynamic approach to the description of economic systems and processes. The first and second laws of thermodynamics as applied to economic systems are derived and analyzed. It is shown that there is a deep analogy between the parameters of thermodynamic and economic systems (markets); in particular, each thermodynamic parameter can be associated with a certain economic parameter or indicator. The economic meaning of such primordial thermodynamic concepts as pressure, volume, internal energy, heat, etc. has been established. The thermostatistics of the market is considered. It is shown that, as in conventional thermostatistics, many market parameters, such as price of goods, quantity of goods, etc., as well as their fluctuations can be calculated formally using the partition function of an economic system. • Economic thermodynamics as a phenomenological theory is developed. • The economic meaning of primordially thermodynamic concepts is established. • The first and second laws of thermodynamics are derived for the economic system. • The thermostatistics of the markets is developed. [ABSTRACT FROM AUTHOR]

We provide the exact solution of several variants of simple models of the zipping transition of two bound polymers, such as occurs in DNA/RNA, in two and three dimensions using pairs of directed lattice paths. In three dimensions the solutions are written in terms of complete Elliptic integrals. In one case the transition occurs at infinite temperature so is less interesting but in other cases occur at finite temperatures. We analyse the phase transition associated in each model giving the scaling of the partition function. We also extend the models to include a pulling force between one end of the pair of paths. • Several variants of models of the zipping transition for two polymers are solved. • These are simple models of DNA/RNA. • Phase diagrams are fully characterised. • First study of a solvable interacting polymer model in three dimensions [ABSTRACT FROM AUTHOR]

We provide a comprehensive computational investigation concerning the effects of confinement and temperature on the thermodynamic properties of cylindrical core/shell quantum dots with a large band offset. This model can also be applied to hollow cylindrical quantum dots or nanofibers. Within the framework of the effective mass approximation, we solve the Schrödinger equation analytically in two bands model, determining the energies of all excited states. Following Boltzmann–Gibbs distribution and introducing the canonical partition function, energy states are used to evaluate the thermodynamic properties: the mean energy, heat capacity, entropy, and Helmholtz free energy. Our numerical calculation shows that all thermodynamic properties depend on the temperature, the size of the dot, and the shell thickness. According to our numerical results, it is found that in the narrow shell case, the heat capacity shows a Schottky-like anomaly at low temperatures, but this effect disappears for small values of core radius. Another important conclusion, is that the determination of the Helmholtz free energy makes it possible to predict the thermodynamic stability of quantum dots. We also show that the competition between the temperature, the core dimension, and the shell thickness influences the thermodynamic stability. Despite the simplicity of our approach, our study can be considered as a useful information source and as an excellent qualitative indicator for understanding the thermodynamic properties of quantum dots. • Determination of thermodynamic properties of SnO 2 /GaAs core/shell quantum dots. • Highlighting the Schottky anomaly of heat capacity in 2D confinement medium. • Comparison of thermodynamic properties of both regular and hollow cylindrical QDots. [ABSTRACT FROM AUTHOR]

It is believed that the canonical gravitational partition function associated with the two-body interacting Newton's gravitation cannot be constructed because the concomitant integral is exponentially divergent. We showed previously that one can indeed obtain finite gravitational results employing both the Gibbs–Boltzmann distribution and Tsallis' one, by recourse to the analytical extension treatment and the generalization of Bollini and Giambiagi's dimensional regularization. We deal here with a model of disc galaxy with a supermassive black hole at its center. Some interesting and coherent results emerge: i—an upper bound in the temperature, ii—the specific heat is negative, iii—the limit of the specific heat when the mass of the black-hole tends to zero is − k B , iv—the third law of thermodynamics is violated, and v—the gravothermal catastrophe is avoided if the number of constituents of a surrounding halo is equal or less than the number of stars in the galaxy. • An upper bound in the temperature of a galaxy is found. • The specific heat is negative. • The limit of the specific heat when the mass of the black-hole tends to zero is − k B . • The third law of thermodynamics is violated. • The gravothermal catastrophe is avoided under some conditions. [ABSTRACT FROM AUTHOR]

Extending the definition of phase-space (Wigner) quantum projectors to thermodynamic ensembles usually results into an efficient platform for quantifying their elementary information content. Given the spectral decomposition profile of a thermalized quantum system, general expressions for the quantum purity quantifier, P (β) , and for phase-space projectors related to the quantum fidelity, F (β) , are explicitly derived in terms of an explicit correspondence with the related partition function, Z (β). Besides quantifying the storage of information capacity of thermodynamical ensembles, the tools here introduced extend the role of the partition function in expressing the quantum behavior of thermodynamic ensembles. • The Weyl–Wigner formalism provides elementary information contents of TD ensembles. • Analytical tools for quantifying the storage of information of TD ensembles are presented. • Expressions for quantum fidelity and purity are explicitly given in terms of the partition function. [ABSTRACT FROM AUTHOR]

This paper explores the use of a cumulant method to determine the zeros of partition functions for continuous phase transitions going up to the second zero. Unlike a first-order transition, with a uniform density of zeros near the transition point, a continuous transition shows a power law dependence of the density with a nontrivial slope for the line of zeros. Different types of models and methods of generating cumulants are used as testing grounds for the method. These include exactly solvable DNA melting problem on hierarchical lattices, heterogeneous DNA melting with randomness in sequence, Monte Carlo simulations for the well-known square lattice Ising model. The method is applicable for zeros near the imaginary axis which control dynamical quantum phase transitions. In all cases, the method is found to provide the basic information about the transition, and most importantly, avoids root finding methods. • Cumulant method to determine two zeros closest to either real or imaginary axis. • Applicability of the cumulant method for data from simulations. • Cumulant method provides a new way of studying disordered systems. • Melting of hetero-DNA has been studied by the cumulant method. • Extraction of critical exponents from the zeros. [ABSTRACT FROM AUTHOR]

In this paper, we consider the thermal properties of one-dimensional Dirac in the framework of the theory of superstatistics where the probability density f (β) follows χ 2 − superstatistics (=Tsallis statistics or Gamma distribution). Under the approximation of the low-energy asymptotics of superstatistics, the partition function, at first, has been calculated by using both Mellin Transformation and Zeta function. This approximation leads to a universal parameter q for any superstatistics, not only for Tsallis statistics. By using the desired partition function, all thermal properties have been obtained in terms of the parameter q. As an application, we extend this concept to the case of Graphene: the reason of this choice is due the existence of an exact mapping about the Dirac oscillator and the compound in question. • Superstatistical Thermal properties of the the one-dimensional Dirac oscillator. • The non-extensive statistics of Tsallis and q-calculs algebra. • Non-extensive statistics of Tsallis and Superstatistics. • Superstatistical Thermal properties of the Graphene. • Mapping between Graphene and Dirac oscillator. [ABSTRACT FROM AUTHOR]