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55 pages, 903 KiB

Open AccessArticle

Random Transitions of a Binary Star in the Canonical Ensemble

by Pierre-Henri Chavanis

Entropy 2024, 26(9), 757; https://doi.org/10.3390/e26090757 - 4 Sep 2024

Viewed by 918

After reviewing the peculiar thermodynamics and statistical mechanics of self-gravitating systems, we consider the case of a “binary star” consisting of two particles of size a in gravitational interaction in a box of radius R. The caloric curve of this system displays a region of negative specific heat in the microcanonical ensemble, which is replaced by a first-order phase transition in the canonical ensemble. The free energy viewed as a thermodynamic potential exhibits two local minima that correspond to two metastable states separated by an unstable maximum forming a barrier of potential. By introducing a Langevin equation to model the interaction of the particles with the thermal bath, we study the random transitions of the system between a “dilute” state, where the particles are well separated, and a “condensed” state, where the particles are bound together. We show that the evolution of the system is given by a Fokker–Planck equation in energy space and that the lifetime of a metastable state is given by the Kramers formula involving the barrier of free energy. This is a particular case of the theory developed in a previous paper (Chavanis, 2005) for N Brownian particles in gravitational interaction associated with the canonical ensemble. In the case of a binary star (

N = 2

), all the quantities can be calculated exactly analytically. We compare these results with those obtained in the mean field limit

N \to + \infty

. Full article

(This article belongs to the Special Issue Statistical Mechanics of Self-Gravitating Systems)

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26 pages, 1605 KiB

Open AccessFeature PaperArticle

Canonical vs. Grand Canonical Ensemble for Bosonic Gases under Harmonic Confinement

by Andrea Crisanti, Luca Salasnich, Alessandro Sarracino and Marco Zannetti

Entropy 2024, 26(5), 367; https://doi.org/10.3390/e26050367 - 26 Apr 2024

Viewed by 1846

Abstract

We analyze the general relation between canonical and grand canonical ensembles in the thermodynamic limit. We begin our discussion by deriving, with an alternative approach, some standard results first obtained by Kac and coworkers in the late 1970s. Then, motivated by the Bose–Einstein condensation (BEC) of trapped gases with a fixed number of atoms, which is well described by the canonical ensemble and by the recent groundbreaking experimental realization of BEC with photons in a dye-filled optical microcavity under genuine grand canonical conditions, we apply our formalism to a system of non-interacting Bose particles confined in a two-dimensional harmonic trap. We discuss in detail the mathematical origin of the inequivalence of ensembles observed in the condensed phase, giving place to the so-called grand canonical catastrophe of density fluctuations. We also provide explicit analytical expressions for the internal energy and specific heat and compare them with available experimental data. For these quantities, we show the equivalence of ensembles in the thermodynamic limit. Full article

(This article belongs to the Special Issue Matter-Aggregating Systems at a Classical vs. Quantum Interface)

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30 pages, 676 KiB

Open AccessArticle

Elasticity of a Grafted Rod-like Filament with Fluctuating Bending Stiffness

by Mohammadhosein Razbin and Panayotis Benetatos

Polymers 2023, 15(10), 2307; https://doi.org/10.3390/polym15102307 - 15 May 2023

Cited by 2 | Viewed by 1387

Abstract

Quite often polymers exhibit different elastic behavior depending on the statistical ensemble (Gibbs vs. Helmholtz). This is an effect of strong fluctuations. In particular, two-state polymers, which locally or globally fluctuate between two classes of microstates, can exhibit strong ensemble inequivalence with negative elastic moduli (extensibility or compressibility) in the Helmholtz ensemble. Two-state polymers consisting of flexible beads and springs have been studied extensively. Recently, similar behavior was predicted in a strongly stretched wormlike chain consisting of a sequence of reversible blocks, fluctuating between two values of the bending stiffness (the so called reversible wormlike chain, rWLC). In this article, we theoretically analyse the elasticity of a grafted rod-like semiflexible filament which fluctuates between two states of bending stiffness. We consider the response to a point force at the fluctuating tip in both the Gibbs and the Helmholtz ensemble. We also calculate the entropic force exerted by the filament on a confining wall. This is done in the Helmholtz ensemble and, under certain conditions, it yields negative compressibility. We consider a two-state homopolymer and a two-block copolymer with two-state blocks. Possible physical realizations of such a system would be grafted DNA or carbon nanorods undergoing hybridization, or grafted F-actin bundles undergoing collective reversible unbinding. Full article

(This article belongs to the Section Polymer Physics and Theory)

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11 pages, 855 KiB

Open AccessArticle

A Convex Hull-Based Machine Learning Algorithm for Multipartite Entanglement Classification

by Pingxun Wang

Appl. Sci. 2022, 12(24), 12778; https://doi.org/10.3390/app122412778 - 13 Dec 2022

Viewed by 1838

Abstract

Quantum entanglement becomes more complicated and capricious when more than two parties are involved. There have been methods for classifying some inequivalent multipartite entanglements, such as GHZ states and W states. In this paper, based on the fact that the set of all W states is convex, we approximate the convex hull by some critical points from the inside and propose a method of classification via the tangent hyperplane. To accelerate the calculation, we bring ensemble learning of machine learning into the algorithm, thus improving the accuracy of the classification. Full article

(This article belongs to the Topic Quantum Information and Quantum Computing)

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19 pages, 394 KiB

Open AccessArticle

Concavity, Response Functions and Replica Energy

by Alessandro Campa, Lapo Casetti, Ivan Latella, Agustín Pérez-Madrid and Stefano Ruffo

Entropy 2018, 20(12), 907; https://doi.org/10.3390/e20120907 - 28 Nov 2018

Cited by 10 | Viewed by 3667

Abstract

In nonadditive systems, like small systems or like long-range interacting systems even in the thermodynamic limit, ensemble inequivalence can be related to the occurrence of negative response functions, this in turn being connected with anomalous concavity properties of the thermodynamic potentials associated with the various ensembles. We show how the type and number of negative response functions depend on which of the quantities E, V and N (energy, volume and number of particles) are constrained in the ensemble. In particular, we consider the unconstrained ensemble in which E, V and N fluctuate, which is physically meaningful only for nonadditive systems. In fact, its partition function is associated with the replica energy, a thermodynamic function that identically vanishes when additivity holds, but that contains relevant information in nonadditive systems. Full article

(This article belongs to the Special Issue Applications of Statistical Thermodynamics)

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17 pages, 339 KiB

Open AccessArticle

Participation Ratio for Constraint-Driven Condensation with Superextensive Mass

by Giacomo Gradenigo and Eric Bertin

Entropy 2017, 19(10), 517; https://doi.org/10.3390/e19100517 - 26 Sep 2017

Cited by 14 | Viewed by 3807

Abstract

Broadly distributed random variables with a power-law distribution

f (m) \sim m^{- (1 + α)}

are known to generate condensation effects. This means that, when the exponent

α

lies in a certain interval, the largest variable in a [...] Read more.

Broadly distributed random variables with a power-law distribution

f (m) \sim m^{- (1 + α)}

are known to generate condensation effects. This means that, when the exponent

α

lies in a certain interval, the largest variable in a sum of N (independent and identically distributed) terms is for large N of the same order as the sum itself. In particular, when the distribution has infinite mean (

0 < α < 1

) one finds unconstrained condensation, whereas for

α > 1

constrained condensation takes places fixing the total mass to a large enough value

M = \sum_{i = 1}^{N} m_{i} > M_{c}

. In both cases, a standard indicator of the condensation phenomenon is the participation ratio

Y_{k} = 〈 \sum_{i} m_{i}^{k} / {(\sum_{i} m_{i})}^{k} 〉

(

k > 1

), which takes a finite value for

N \to \infty

when condensation occurs. To better understand the connection between constrained and unconstrained condensation, we study here the situation when the total mass is fixed to a superextensive value

M \sim N^{1 + δ}

(

δ > 0

), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as

M \sim N^{1 / α}

for

α < 1

) and the extensive constrained mass. In particular we show that for exponents

α < 1

a condensate phase for values

δ > δ_{c} = 1 / α - 1

is separated from a homogeneous phase at

δ < δ_{c}

from a transition line,

δ = δ_{c}

, where a weak condensation phenomenon takes place. We focus on the evaluation of the participation ratio as a generic indicator of condensation, also recalling or presenting results in the standard cases of unconstrained mass and of fixed extensive mass. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

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