Chapter 17 - Canonical ensemble

Chapter 10 - Distribution of independent particles

This chapter discusses the canonical partition function (Q). It notes that the molecular partition function contains all the thermodynamic information about a system of independent particles at equilibrium. Additionally, the chapter highlights the molecular partition function as being the number of energy states accessible to the system at a given temperature. The chapter notes the system wherein energy states are no longer constant and fluctuate over time. The canonical partition function Q does not rely on the premise of individual particles interacting with each other. Additionally, Q provides a major tool that can be used when looking at increasingly complex situations such as understanding the relationship between the molecular partition function and the canonical partition function.

Canonical partition functions and Lee-Yang zeros of QCD at finite density and high temperature are studied. Recent lattice simulations have confirmed that the free energy of QCD is a quartic function of quark chemical potential at temperature slightly above pseudo-critical temperature $T_c$, as in the case with a gas of free massless fermions. We present analytic derivation of the canonical partition functions and Lee-Yang zeros for this type of free energy using the saddle point approximation. We also perform lattice QCD simulation in a canonical approach using the fugacity expansion of the fermion determinant, and carefully examine its reliability. By comparing the analytic and numerical results, we conclude that the canonical partition functions follow the Gaussian distribution of the baryon number, and the accumulation of Lee-Yang zeros of these canonical partition functions exhibit the first-order Roberge-Weiss phase transition. We discuss the validity and applicable range of the result and its implications both for theoretical and experimental studies.

Statistical mechanics has a universal appeal that extends beyond molecular systems, and yet, as its tools are being transplanted to fields outside physics, the fundamental questions, what is thermodynamics and how it may be applied outside the realm of physical particles, have remained unanswered. We answer these questions here: Statistical mechanics in its most general form is variational calculus applied to probability distributions and by extension to stochastic processes in general; as a mathematical theory, it is independent of physical hypotheses but provides the means to incorporate our knowledge and model assumptions about the particular problem. The fundamental ensemble is a microcanonical space of probability distributions sampled via a bias functional that establishes a probability measure on this space. The maximization of this measure expresses the most probable distribution via a set of parameters (microcanonical partition function, canonical partition function and generalized temperature) that are connected through a set of mathematical relationships that we recognize as the familiar equations of thermodynamic. Any distribution in in this space maybe endowed with the status of the most probable distribution under an appropriately constructed bias functional. Entropy, Kullback-Leibler divergence and the second law have simple interpretations in this theory. We obtain statistical mechanics as a special application to molecular systems and make contact with Information Theory and Bayesian inference. We use numerical examples to demonstrate the thermodynamic treatment of generic probability distributions, present a thermodynamic algorithm (the cluster ensemble) to sample arbitrary distributions with positive argument by analogy to reacting particles and discuss the extension of statistical mechanics to stochastic processes in general.

This chapter explores more powerful methods of calculation than were seen previously. Among them are Molecular Dynamics (MD) and Monte Carlo (MC) computer simulations. Another is the canonical partition function, which is related to the Helmholtz free energy. The derivation of thermodynamic identities within statistical mechanics is illustrated by the relationship between the specific heat and the fluctuations of the energy. It is shown how the canonical ensemble allows us to integrate out the momentum variables for many classical models. The factorization of the partition function is presented as the best trick in statistical mechanics, because of its central role in solving problems. Finally, the problem of many simple harmonic oscillators is solved, both for its importance and as an illustration of the best trick.

A new lattice model for nematic and/or smectic liquid crystals with rigid (inflexible) molecular cores and semiflexible pendent tails is presented and an approximate expression for its canonical ensemble partition function is derived. The model system consists of N rodlike molecules on a face-centered-cubic (fcc) lattice of M sites; each molecule has a rigid central core of r segments and two semiflexible tails of f/2 segments each and occupies x=r+f contiguous sites on the lattice. Molecular cores are restricted to three mutually orthogonal orientations while each tail bond can have five possible orientations with respect to the preceding bond, corresponding (roughly) to five of the nine possible rotational-isomeric states of a pair of successive C-C bonds on a saturated molecular tail. In addition to hard-core exclusions, there are interactions between pairs of segments on different molecules separated by the first-, second-, and third-nearest-neighbor distances on the fcc lattice, and a positive energy is required to make each bend in a molecular tail. The canonical ensemble partition function is derived using the Bragg-Williams approximation to treat intermolecular interactions (other than hard-core exclusions) and an extension of the DiMarzio counting scheme to evaluate the number of ways to pack the molecules on the lattice. This model and statistical-mechanical approach are improvements over earlier lattice-model treatments of nematic and smectic liquid-crystal-forming substances in the following major respects: (1) The treatment of smectic positional order is much less approximate than the earlier treatment of Dowell. (2) The representation of the rotational-isomeric states of a molecular tail is considerably more realistic than was possible with previous lattice models. (3) Positional, orientational, and conformational order are all coupled. (4) The conformational statistics of the model molecular tails are treated exactly within the maximum-term approximation.

Study of canonical entropy in electron-proton interactions at s=300 GeV is presented. The precision data collected by the H1 experiment at the HERA in different ranges of invariant hadronic mass W and the squared four-momentum exchange Q2 in electron-proton (ep) interactions have been analyzed in the ensemble theory approach. The canonical partition function relates to the multiplicity distribution which is often studied in collider experiments. We use the canonical ensemble partition function to explore the dynamics of hadron production in ep interactions by devising different methods to find the entropic parameter and the collision temperature. The inverse slope of the transverse momentum spectrum of produced hadrons also relates to the temperature. In the recent past, the CMS, ATLAS, and ALICE experiments at the LHC have studied the charged hadron transverse momentum and particle distributions in proton-proton and proton-nucleus interactions by using the Tsallis function within this approach. A detailed investigation into the role of the system volume and relation amongst different dynamical parameters reveals interesting results. Published by the American Physical Society 2024

Supramolecular structure design by computer simulations can be an effective method in molecular engineering of surfaces using self-assembled monolayers. In this contribution we describe the use of the Monte Carlo simulation technique for the two-dimensional self-organization of model tripod molecules on a solid surface. To that end binary mixtures of flat symmetric molecules adsorbed on a triangular lattice were simulated using the Canonical Ensemble method. Special attention was paid to the influence of the difference between sizes of the components on the formation of highly ordered superstructures. It was demonstrated that the tripod molecules having sufficiently small size can co-assemble into binary hierarchical networks whose structural properties can be finely tuned by changing the composition of the mixture. For those networks the chiral occlusion effect was observed, in which molecules of smaller component are confined to form chiral pores. The insights from the simulations can be helpful in custom designing of chiral porous networks in two dimensions, as they establish a link between structural properties of the building blocks and morphology of the resulting adlayer.

In this work, thermodynamic properties of the three-dimensional charged dilatonic black holes are considered in the presence of Born-Infeld nonlinear electrodynamics. The exact nonlinearly charged black hole solutions to the coupled Einstein-Born-Infeld three-dimensional field equations in the presence of a dilatonic scalar field are constructed. Some new classes of nonlinearly charged dilatonic black hole solutions are distinguished according to different values of the parameters in the theory. The thermodynamics of all of the new Einstein-Born-Infeld-dilaton black holes are studied separately. We show that the thermodynamic quantities satisfy the first law of black hole thermodynamics. Through the canonical ensemble method and noting the black hole heat capacity, we analyze the stability and thermodynamic phase transitions of all of the new Einstein-Born-Infeld-dilaton black holes.

We explore three-dimensional scalar-tensor black hole solutions in the presence of power-Maxwell electrodynamics. By applying the conformal transformations on the action of scalar-tensor theory, we show that the electromagnetic Lagrangian remains invariant for a specific amount of power. In addition, the gravitational action transforms to that of Einstein-dilaton gravity theory. Through solving the field equations of this theory, we obtain two novel classes of Einstein-dilaton black hole solutions. Next, proceed to investigate the thermodynamic properties of the solutions and show that thermodynamical first law is valid for both of our solutions. Then, we analyze the thermal stability or phase transition of the black holes based on the canonical ensemble method. Finally, we obtain the Jordan frame scalar-tensor black hole solutions, by imposing the inverse conformal transformations, from their Einstein-dilaton counterparts and investigate their thermodynamics and thermal stability properties.

For many-particle quantum systems, calculating thermodynamic quantities in the canonical ensemble is a very hard task, while this is tractable in the grand canonical ensemble.The second ensemble is then used.The results are supposed to be the same, at least in the thermodynamic limit.Is this actually the case?In this work, we consider a system of N noninteracting bosons distributed among few energy levels.We can calculate the canonical partition function in this case and deduce the canonical mean energy.We compare it to the mean energy deduced from the grand canonical ensemble for the same number of particles.We consider the case of a large number of particles.

We propose a simple pedagogical way of introducing the Euler–MacLaurin summation formula in an undergraduate course on statistical mechanics. The reason is that the students may feel more comfortable and confident if they are able to deduce the main equations. To this end we put forward two alternative routes: the first one is the simplest and yields the first two terms of the expansion. The second one is somewhat more elaborate and takes into account all the correction terms. We apply both to the calculation of the simplest one-particle canonical partition functions for the translational, vibrational and rotational degrees of freedom. The more elaborate, systematic calculation of the correction terms is suitable for motivating the students to explore the possibility of using available computer algebra software that enable one to avoid long and tedious manipulation of algebraic equations.

Canonical Bose-Einstein condensation of interacting bosons in two dimensions

The probability distribution of a function of a subsystem conditioned on the value of the function of the whole, in the limit when the ratio of their values goes to zero, has a limit law: It equals the unconditioned marginal probability distribution weighted by an exponential factor whose exponent is uniquely determined by the condition. We apply this theorem to explain the canonical equilibrium ensemble of a system in contact with a heat reservoir. Since the theorem only requires analysis at the level of the function of the subsystem and reservoir, it is applicable even without the knowledge of the composition of the reservoir itself, which extends the applicability of the canonical ensemble. Furthermore, we generalize our theorem to a model with strong interaction that contributes an additional term to the exponent, which is beyond the typical case of approximately additive functions. This result is new in both physics and mathematics, as a theory for the Gibbs conditioning principle for strongly correlated systems. A corollary provides a precise formulation of what a temperature bath is in probabilistic term

We introduce ensembles of repelling charged particles restricted to a ball in a non-archimedean field (such as the p p -adic rational numbers) with interaction energy between pairs of particles proportional to the logarithm of the ( p p -adic) distance between them. In the canonical ensemble , a system of N N particles is put in contact with a heat bath at fixed inverse temperature β \beta and energy is allowed to flow between the system and the heat bath. Using standard axioms of statistical physics, the relative density of states is given by the β \beta power of the ( p p -adic) absolute value of the Vandermonde determinant in the locations of the particles. The partition function is the normalizing constant (as a function of β \beta ) of this ensemble, and we identify a recursion that allows this to be computed explicitly in finite time. Probabilities of interest, including the probabilities that fixed subsets will have a prescribed number of particles, and the conditional distribution of particles within a subset given a prescribed occupation number, are given explicitly in terms of the partition function. We then turn to the grand canonical ensemble where both the energy and number of particles are variable. We compute similar probabilities to those in the canonical ensemble and show how these probabilities can be given in terms the canonical and grand canonical partition functions. Finally, we briefly consider the multi-component ensemble where particles are allowed to take different integer charges, and we connect basic properties of this ensemble to the canonical and grand canonical ensembles.