Non-vanishing Entropy Research Articles

Ensemble nonequivalence and Bose–Einstein condensation in weighted networks

The asymptotic (non)equivalence of canonical and microcanonical ensembles, describing systems with soft and hard constraints respectively, is a central concept in statistical physics. Traditionally, the breakdown of ensemble equivalence (EE) has been associated with nonvanishing relative canonical fluctuations of the constraints in the thermodynamic limit. Recently, it has been reformulated in terms of a nonvanishing relative entropy density between microcanonical and canonical probabilities. The earliest observations of EE violation required phase transitions or long-range interactions. More recent research on binary networks found that an extensive number of local constraints can also break EE, even in absence of phase transitions. Here we study for the first time ensemble nonequivalence in weighted networks with local constraints. Unlike their binary counterparts, these networks can undergo a form of Bose–Einstein condensation (BEC) producing a core–periphery structure where a finite fraction of the link weights concentrates in the core. This phenomenon creates a unique setting where local constraints coexist with a phase transition. We find surviving relative fluctuations only in the condensed phase, as in more traditional BEC settings. However, we also find a non-vanishing relative entropy density for all temperatures, signalling a breakdown of EE due to the presence of an extensive number of constraints, irrespective of BEC. Therefore, in presence of extensively many local constraints, vanishing relative fluctuations no longer guarantee EE.

Inferring entropy production in anharmonic Brownian gyrators

A non-vanishing entropy production rate is one of the defining characteristics of any non-equilibrium system, and several techniques exist to determine this quantity directly from experimental data. The short-time inference scheme, derived from the thermodynamic uncertainty relation, is a recent addition to the list of these techniques. Here we apply this scheme to quantify the entropy production rate in a class of microscopic heat engine models called Brownian gyrators. In particular, we consider models with anharmonic confining potentials. In these cases, the dynamical equations are indelibly non-linear, and the exact dependences of the entropy production rate on the model parameters are unknown. Our results demonstrate that the short-time inference scheme can efficiently determine these dependencies from a moderate amount of trajectory data. Furthermore, the results show that the non-equilibrium properties of the gyrator model with anharmonic confining potentials are considerably different from its harmonic counterpart - especially in set-ups leading to a non-equilibrium dynamics and the resulting gyration patterns.

Entanglement entropy of inhomogeneous XX spin chains with algebraic interactions

We introduce a family of inhomogeneous XX spin chains whose squared couplings are a polynomial of degree at most four in the site index. We show how to obtain an asymptotic approximation for the Rényi entanglement entropy of all such chains in a constant magnetic field at half filling by exploiting their connection with the conformal field theory of a massless Dirac fermion in a suitably curved static background. We study the above approximation for three particular chains in the family, two of them related to well-known quasi-exactly solvable quantum models on the line and the third one to classical Krawtchouk polynomials, finding an excellent agreement with the exact value obtained numerically when the Rényi parameter α is less than one. When α ≥ 1 we find parity oscillations, as expected from the homogeneous case, and show that they are very accurately reproduced by a modification of the Fagotti-Calabrese formula. We have also analyzed the asymptotic behavior of the Rényi entanglement entropy in the non-standard situation of arbitrary filling and/or inhomogeneous magnetic field. Our numerical results show that in this case a block of spins at each end of the chain becomes disentangled from the rest. Moreover, the asymptotic approximation for the case of half filling and constant magnetic field, when suitably rescaled to the region of non-vanishing entropy, provides a rough approximation to the entanglement entropy also in this general case.

Second order relativistic viscous hydrodynamics within an effective description of hot QCD medium

The second-order hydrodynamic equations for evolution of shear and bulk viscous pressure have been derived within the framework of covariant kinetic theory based on the effective fugacity quasiparticle model. The temperature-dependent fugacity parameter in the equilibrium distribution function leads to a mean field term in the Boltzmann equation which affects the interactions in the hot QCD matter. The viscous corrections to distribution function, up to second-order in gradient expansion, have been obtained by employing a Chapman–Enskog like iterative solution of the effective Boltzmann equation within the relaxation time approximation. The effect of mean field contributions to transport coefficients as well as entropy current has been studied up to second-order in gradients. In contrast to the previous calculations, we find non-vanishing entropy flux at second order. The effective description of relativistic second-order viscous hydrodynamics, for a system of interacting quarks and gluons, has been quantitatively analyzed in the case of the 1 + 1-dimensional boost invariant longitudinal expansion. We study the proper time evolution of temperature, pressure anisotropy, and viscous corrections to entropy density for this simplified expansion. The second order evolution of quark–gluon plasma is seen to be affected significantly with the inclusion of mean field contributions and the realistic equation of state.

Silence of Binary Kerr Black Holes.

A nontrivial S matrix generally implies a production of entanglement: starting with an incoming pure state, the scattering generally returns an outgoing state with nonvanishing entanglement entropy. It is then interesting to ask if there exists a nontrivial S matrix that generates no entanglement. In this Letter, we argue that the answer is the S-matrix for the scattering of classical black holes. We study the spin entanglement in the scattering of arbitrary spinning particles. Augmenting the S-matrix with Thomas-Wigner rotation factors, we derive the entanglement entropy from the gravitational induced 2→2 amplitude. In the Eikonal limit, we find that the relative entanglement entropy, defined here as the difference between the entanglement entropy of the in and out states, is nearly zero for minimal coupling irrespective of the in state and increases significantly for any nonvanishing spin multipole moments. This finding suggests that minimal couplings of spinning particles, whose classical limit corresponds to a Kerr black hole, have the unique feature of generating near zero entanglement.

Entanglement entropy of cosmological perturbations

We show that the entropy of cosmological perturbations originating as quantum vacuum fluctuations in the very early universe, including the contribution of the leading nonlinear interactions, can be viewed as momentum space entanglement entropy between sub- and super-Hubble modes. The interactions between these modes causes decoherence of the super-Hubble fluctuations which, in turn, leads to a non-vanishing entropy of the reduced density matrix corresponding to the super-Hubble inhomogeneities. In particular, applying this to inflationary cosmology reveals that the entanglement entropy produced by leading order nonlinearities dominates over that coming from the squeezing of the vacuum state unless inflation lasts for a very short period. Furthermore, demanding that this entanglement entropy be smaller than the thermal entropy at the beginning of the radiation phase of standard cosmology leads to an upper bound on the duration of inflation which is similar to what is obtained from the Trans-Planckian Censorship Conjecture.

Thermoelectric power of Sachdev-Ye-Kitaev islands: Probing Bekenstein-Hawking entropy in quantum matter experiments

The Sachdev-Ye-Kitaev (SYK) model describes electrons with random and all-to-all interactions, and realizes a many-body state without quasiparticle excitations, and a non-vanishing extensive entropy $S_0$ in the zero temperature limit. Its low energy theory coincides the low energy theory of the near-extremal charged black holes with Bekenstein-Hawking entropy $S_0$. Several mesoscopic experimental configurations realizing SYK quantum dynamics over a significant intermediate temperature scale have been proposed. We investigate quantum thermoelectric transport in such configurations, and describe the low temperature crossovers out of SYK criticality into regimes with either Fermi liquid behavior, a Coulomb blockade, or criticality associated with Schwarzian quantum gravity fluctuations. Our results show that thermopower measurements can serve as a direct probe for $S_0$.

Entropy Dynamics of Phonon Quantum States Generated by Optical Excitation of a Two-Level System.

In quantum physics, two prototypical model systems stand out due to their wide range of applications. These are the two-level system (TLS) and the harmonic oscillator. The former is often an ideal model for confined charge or spin systems and the latter for lattice vibrations, i.e., phonons. Here, we couple these two systems, which leads to numerous fascinating physical phenomena. Practically, we consider different optical excitations and decay scenarios of a TLS, focusing on the generated dynamics of a single phonon mode that couples to the TLS. Special emphasis is placed on the entropy of the different parts of the system, predominantly the phonons. While, without any decay, the entire system is always in a pure state, resulting in a vanishing entropy, the complex interplay between the single parts results in non-vanishing respective entanglement entropies and non-trivial dynamics of them. Taking a decay of the TLS into account leads to a non-vanishing entropy of the full system and additional aspects in its dynamics. We demonstrate that all aspects of the entropy’s behavior can be traced back to the purity of the states and are illustrated by phonon Wigner functions in phase space.

Quantum many-body conformal dynamics: Symmetries, geometry, conformal tower states, and entropy production

In this article we study the quench dynamics of Galilean and scale invariant many-body systems which can be prepared using interacting atomic gases. The far-away from equilibrium dynamics are investigated by employing $m$-body density matrices, which are most conveniently defined in terms of a special basis - the conformal tower states. We explicitly illustrate that, although during the initial stage of the dynamics all symmetries can be broken and absent in the unitary evolution because of the initialization of the state, there is always an emergent conformal symmetry in the long time limit. The emergence of this dynamic conformal symmetry is robust, and always occurs; it uniquely defines the characteristics of the asymptotic dynamics at a scale invariant fixed point. As an immediate application of the asymptotic dynamics of the microscopic density matrices, we have focused on the effects of this emergent conformal symmetry on two observables: the moment of inertia tensor, $I_{ij}(t)$, $i,j=x,y,z$, and the entropy density field, $S({\bf r}, t)$, in the hydrodynamic flow of strongly interacting particles. We show that the long time behaviour of these observables is completely set by conformal symmetry, while the leading long time corrections depend on interference effects between different conformal tower states. The emergent conformal symmetry naturally leads to entropy conservation, and {\em conformal cooling}, an energy conserving cooling of a strongly interacting gas during free expansion. When the interaction Hamiltonian breaks the scale symmetry, we further demonstrate that there is a direct cause-effect relation between conformal symmetry breaking in the long time limit, and a non-vanishing entropy production. This suggests that the entropy production rate is a natural {\em parameter} for categorizing the breaking of conformal symmetry.

Entangled de Sitter from stringy axionic Bell pair I: an analysis using Bunch\u2013Davies vacuum

In this work, we study the quantum entanglement and compute entanglement entropy in de Sitter space for a bipartite quantum field theory driven by an axion originating from Type IIB string compactification on a Calabi–Yau three fold (mathbf{CY^3}) and in the presence of an mathbf{NS5} brane. For this computation, we consider a spherical surface mathbf{S}^2, which divides the spatial slice of de Sitter (mathbf{dS_4}) into exterior and interior sub-regions. We also consider the initial choice of vacuum to be Bunch–Davies state. First we derive the solution of the wave function of the axion in a hyperbolic open chart by constructing a suitable basis for Bunch–Davies vacuum state using Bogoliubov transformation. We then derive the expression for density matrix by tracing over the exterior region. This allows us to compute the entanglement entropy and Rényi entropy in 3+1 dimension. Furthermore, we quantify the UV-finite contribution of the entanglement entropy which contain the physics of long range quantum correlations of our expanding universe. Finally, our analysis complements the necessary condition for generating non-vanishing entanglement entropy in primordial cosmology due to the axion.

Degenerated ground-states in a spin chain with pair interactions: a characterization by symbolic dynamics

In this paper we study a class of one-dimensional spin chain having a highly degenerated set of ground-state configurations. The model consists of spin chain having infinite-range pair interactions with a given structure. We show that the set of ground-state configurations of such a model can be fully characterized by means of symbolic dynamics. Particularly we found that the set ground-state configurations define what in symbolic dynamics is called sofic shift space. Finally we prove that this system has a non-vanishing residual entropy (the topological entropy of the shift space), which can be exactly calculated.

Thermodynamic formalism and linear response theory for nonequilibrium steady states.

We study the linear response in systems driven away from thermal equilibrium into a nonequilibrium steady state with nonvanishing entropy production rate. A simple derivation of a general response formula is presented under the condition that the generating function describes a transformation that (to lowest order) preserves normalization and thus describes a physical stochastic process. For Markov processes we explicitly construct the conjugate quantities and discuss their relation with known response formulas. Emphasis is put on the formal analogy with thermodynamic potentials and some consequences are discussed.

The third law of thermodynamics and the fractional entropies

We consider the fractal calculus based Ubriaco and Machado entropies and investigate whether they conform to the third law of thermodynamics. The Ubriaco entropy satisfies the third law of thermodynamics in the interval 0<q≤1 exactly where it is also thermodynamically stable. The Machado entropy, on the other hand, yields diverging inverse temperature in the region 0<q≤1, albeit with non-vanishing negative entropy values. Therefore, despite the divergent inverse temperature behavior, the Machado entropy fails the third law of thermodynamics. We also show that the aforementioned results are also supported by the one-dimensional Ising model with no external field.

Entropy of regular timed languages

To study the size of regular timed languages, we generalize a classical approach introduced by Chomsky and Miller for discrete automata: count words having n symbols, and compute the exponential growth rate of their number (entropy). For timed automata, we replace cardinality by volume and define (volumetric) entropy similarly. It represents the average quantity of information per event in a timed word of the language. We exhibit a criterion for telling apart “thick” timed automata with non-vanishing entropy, for which typical runs are non-Zeno and discretizable, from “thin” automata for which all runs behave in a Zeno-like way, implying a quick volume collapse. We associate to every timed automaton a positive integral operator; the entropy equals the logarithm of its spectral radius. This operator has a spectral gap, thus allowing for fast converging numerical procedures to approximate entropy. In a special case, entropy is even characterized symbolically.

Relativistic third-order viscous corrections to the entropy four-current from kinetic theory

By employing a Chapman-Enskog like iterative solution of the Boltzmann equation in relaxation-time approximation, we derive a new expression for the entropy four-current up to third order in gradient expansion. We show that unlike second-order and third-order entropy four-current obtained using Grad's method, there is a nonvanishing entropy flux in the present third-order expression. We further quantify the effect of the higher-order entropy density in the case of boost-invariant one-dimensional longitudinal expansion of a system. We demonstrate that the results obtained using the third-order evolution equation for the shear stress tensor, derived by employing the method of Chapman-Enskog expansion, show better agreement with the exact solution of the Boltzmann equation as well as with the parton cascade bamps, as compared to those obtained using the third-order equations from the method of Grad's 14-moment approximation.

Extremal black holes and the first law of thermodynamics

We study the low temperature expansion of the first law of thermodynamics for near-extremal black holes. We show that for extremal black holes with non-vanishing entropy, the leading order contribution yields an expression for their extremal entropy in agreement with the entropy function result and the Cardy formula for the entropy of a two dimensional chiral conformal field theory (CFT). When their entropy vanishes due to the vanishing of a one-cycle on the horizon, such leading contribution is always compatible with the first law satisfied by a BTZ black hole. These results are universal and consistent both with the presence of local AdS2 and AdS3 near horizon throats for extremal black holes and with the suggested quantum microscopic descriptions (AdS2/CFT1, Kerr/CFT and EVH/CFT).

Exact correlation functions in particle-reaction models with immobile particles

Exact results on particle densities as well as correlators in two models of immobile particles, containing either a single species or else two distinct species, are derived. The models evolve following a descent dynamics through pair annihilation where each particle interacts once at most throughout its entire history. The resulting large number of stationary states leads to a non-vanishing configurational entropy. Our results are established for arbitrary initial conditions and are derived via a generating function method. The single-species model is the dual of the 1D zero-temperature kinetic Ising model with Kimball–Deker–Haake dynamics. In this way, both infinite and semi-infinite chains and also the Bethe lattice can be analysed. The relationship with the random sequential adsorption of dimers and weakly tapped granular materials is discussed.

Vorticity and entropy production in tilted Szekeres spacetimes

We analyze the properties of the tilted Szekeres spacetime, i.e. the version of such spacetime as seen by a congruence of observers with respect to which the fluid is moving. The imperfect fluid and the kinematical variables associated to the four-velocity of the fluid assigned by tilted observers are studied in detail. As it happens for the case of the Lemaitre--Tolman--Bondi spacetime, the fluid evolves nonreversibly (with nonvanishing entropy production) and is nongeodesic. However unlike that later case, the tilted observer detects vorticity in the congruence of the fluid world lines. Also, as for the nontilted congruence the magnetic part of the Weyl tensor vanishes, reinforcing the nonradiative character of this kind of spacetime. Possible physical implications of these results are discussed.

Entangled states and entropy remnants of a photon–electron system

In the present paper, an example of entanglement between two different kinds of interacting particles, photons and electrons is analysed. The initial-value problem of the Schrödinger equation is solved non-perturbatively for the system of a free electron interacting with a quantized mode of electromagnetic radiation. Wave packets of the dressed states so obtained are constructed in order to describe the spatio-temporal separation of the subsystems before and after interaction. The joint probability amplitudes are calculated for the detection of the electron at some space–time location and the detection of a definite number of photons. The analytical study of the time evolution of entanglement between the initially separated electron wave packet and the radiation mode leads to the conclusion that in general there are non-vanishing entropy remnants in the subsystems after the interaction. On the basis of the simple model to be presented here, the calculated values of the entropy remnants crucially depend on the character of the assumed switching-on and -off of the interaction.

Semi-global simulations of the magneto-rotational instability in core collapse supernovae

Context. Possible effects of magnetic fields in core collapse supernovae rely on an efficient amplification of the weak pre-collapse fields. It has been suggested that the magneto-rotational instability (MRI) leads to a rapid growth for these weak seed fields. Although plenty of MRI studies exist for accretion disks, the application of their results to core collapse supernovae is inhibited as the physics of supernova cores is substantially different from that of accretion discs. Aims. We address the problem of growth and saturation of the MRI in core collapse supernovae by studying its evolution by means of semi-global simulations, which combine elements of global and local simulations by taking the presence of global background gradients into account and using a local computational grid. We investigate, in particular, the termination of the growth of the MRI and the properties of the turbulence in the saturated state.Methods. We analyze the dispersion relation of the MRI to identify different regimes of the instability. This analysis is complemented by semi-global ideal MHD simulations, where we consider core matter in a local computational box (size ∼ km) rotating at sub-Keplerian velocity and where we allow for the presence of a radial entropy gradient, but neglect neutrino radiation. Results. We identify six regimes of the MRI depending on the ratio of the entropy and angular velocity gradient. Our numerical models confirm the instability criteria and growth rates for all regimes relevant to core-collapse supernovae. The MRI grows exponentially on time scales of milliseconds, the flow and magnetic field geometries being dominated by channel flows. We find MHD turbulence and efficient transport of angular momentum. The MRI growth ceases once the channels are disrupted by resistive instabilities (stemming from to the finite conductivity of the numerical code), and MHD turbulence sets in. From an analysis of the growth rates of the resistive instabilities, we deduce scaling laws for the termination amplitude of the MRI, which agree well with our numerical models. We determine the dependence of the development of large-scale coherent flow structures in the saturated state on the aspect ratio of the simulation boxes. Conclusions. The MRI can grow rapidly under the conditions considered here, i.e., a rapidly rotating core in hydrostatic equilibrium, possibly endowed with a nonvanishing entropy gradient, leading to fields exceeding . More investigations are required to cover the parameter space more comprehensively and to include more physical effects.