# Statistical Thermodynamics Fundamentals An

In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles.

## Statistical Thermodynamics Fundamentals an

Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinate axes. In quantum statistical mechanics, the ensemble is a probability distribution over pure states,[note 1] and can be compactly summarized as a density matrix.

One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state.[note 2] The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.

The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material.

Whereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving (mechanical equilibrium), rather, only that the ensemble is not evolving.

A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).[1]There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics.[1] Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.

Other fundamental postulates for statistical mechanics have also been proposed.[4][5][6] For example, recent studies shows that the theory of statistical mechanics can be built without the equal a priori probability postulate.[5][6] One such formalism is based on the fundamental thermodynamic relation together with the following set of postulates:[5]

There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume.[1] These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.

All of these processes occur over time with characteristic rates. These rates are important in engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)

In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.

One approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.

The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the "interesting" information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors), where the electrons are indeed analogous to a rarefied gas.

Analytical and computational techniques derived from statistical physics of disordered systems, can be extended to large-scale problems, including machine learning, e.g., to analyze the weight space of deep neural networks.[15] Statistical physics is thus finding applications in the area of medical diagnostics.[16]

In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range.[17] This was the first-ever statistical law in physics.[18] Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium.[19] Five years later, in 1864, Ludwig Boltzmann, a young student in Vienna, came across Maxwell's paper and spent much of his life developing the subject further.

Statistical mechanics was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory.[20] Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.

In equilibrium statistical physics the fundamental assumption of statistical thermodynamics states that the occupation of any microstate is equally probable (i.e. $p_i=1/\Omega, S=-k_B\sum p_i\,\rm ln\,p_i=k_B\rm ln\,\Omega$). But for isolated system in equilibrium we also have Boltzmann distribution which states $p_i=e^-\beta E_i/Z$, where $E_i$ are the allowed energy levels. So the two $p_i$ matches if and only if there is one single allowed energy level. How can we resolve this conflict?

With the Boltzmann distribution (AKA canonical ensemble) this assumption doesn't apply since we have knowledge about the system. In particular we know that if the system is put into contact with a thermodynamic heat bath of temperature $T$ (the same $T$ as in the $e^-E/(kT)$ distribution), the system will remain in statistical equilibrium (the distribution will not change). This property, of being in equilibrium with other systems of the same temperature, is special to the Boltzmann distribution and is what makes it so useful.

Advanced topics in macroscopic thermodynamics and fundamentals of statistical thermodynamics. Thermodynamics of composite systems including surface thermodynamics and thermodynamics in fields. Introduction to quantum mechanics. Principles of statistical thermodynamics. Construction of partition functions and calculations of basic thermodynamic properties for several fundamental systems. Applications will include properties of ideal gases, ideal solids and adsorbed gases.

Description: The discussion on statistical physics continues in this lecture. The instructor gives several examples in different ensemble cases, and also an application example in gas molecule.

Thermodynamics is the foundation on which the science of physical chemistry is built, and statistical thermodynamics provides the fundamental, molecular-level basis for the ideas of thermodynamics. This course will focus on the ability of statistical thermodynamics to employ simple physical models, along with some inspired mathematics, to predict the behavior of atoms and molecules (referred to as, "the unreasonable effectiveness of unrealistic simplifications"). The concept of entropy will be a unifying theme throughout the course, and, given its central role, time will be devoted to developing a rigorous mathematical model of entropy through the use of probability and multi-variable calculus. The more traditional topics of thermodynamics will be presented relatively quickly. The first and second laws of thermodynamics will be explored; the fundamental equations of thermodynamics, as differential equations, will be used to define the properties of temperature, pressure and chemical potential; and the concept of free energy (and its importance in describing equilibrium) will be developed. The Boltzmann distribution law (and the partition function) will be derived and then used, along with simple physical models, to compute thermodynamic and physical properties of systems at equilibrium. Following a brief look at quantum theory and statistical mechanics, the equilibrium constant expression will be derived (by employing the partition function) and values for gas-phase chemical equilibrium constants will be computed and compared to empirical values. Finally, lattice models will be used to explore properties of liquids, liquid/vapor equilibrium and solutions. There will be some experimental work, allowing students the opportunity to study an actual system and use the simple models, and mathematics, of statistical thermodynamics to investigate its physical and/or chemical behavior. 041b061a72