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Statistical Physics

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Statistical mechanics aims at studying the macroscopic parameters of a system in equilibrium from a knowledge of the microscopic properties of its constituent particles using the laws of mechanics. This is different from the approach of thermodynamics which aims at studying a macroscopic system in equilibrium from macroscopic phenomenological standpoint without considering the microscopic constituents of the system.  

 

Definition

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Statistical physics is the branch of physics where a scheme to calculate the free energy is formulated. In statistical physics, we use the fact that matter consists of atoms. On the basis of a knowledge of the microscopic laws that govern the motion of atoms and most importantly an additional law of statistical physics, it gives a general expression for the free energy. Since statistical physics starts from the microscopic level, it can discuss not only thermal equilibrium states, but also non equilibrium states. Small deviations from thermal equilibrium can be discussed by the use of linear response theory, and we can discuss such effects as electrical or thermal conductivity. However, the statistical physics of non equilibrium states is not yet well established, especially for states far from equilibrium. It is an actively investigated branch of physics even today.  

Statistical Mechanics

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In statistical mechanics we study the physical systems consisting of very large number of particles. The simplest system of interest is a perfect gas in thermal equilibrium. From the macroscopic point of view it appears to be a continuum. A complete set of thermodynamic variables, characterizing its equilibrium state, is the energy E, volume V and the number of molecules N. The N, although referred to molecules for convenience, is a macroscopic variable because it is directly related to the mass of the gas. From the microscopic point of view, the gas (matter) consists of discrete particles, like atoms or molecules. 

Thermodynamics and Statistical Mechanics

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Thermal physics or thermodynamics, treats a macroscopic sample of matter such as a gas or a solid as a black box. It provides general laws for the response of matter to actions from the environment. For instance, when we exert a force on a volume of gas in the form of a pressure, the gas will contract. When we give energy in the form of a pressure, the gas either expand or increase its pressure. There are general relationships between these responses. Thermodynamics gives us such relationships. This branch of physics evolved from the necessity to increase the efficiency of the conversion of heat to work, which became important after the industrial revolution. The laws of thermodynamics are quite general; they are independent of the species of the atoms from which the matter is constructed and independent of the interactions between the atoms. 

Statistical mechanics as Balescu states is the "mechanics of large assemblies of simple systems such as molecules in a gas, atoms in a crystal". In chemistry we are interested in the prediction of macroscopic (bulk) properties such as pressure and density which are obviously related to the properties of individual atoms and molecules. The purpose of statistical mechanics is "to understand the behavior of the assembly as a whole in terms of the behavior of its constituents" using statistical considerations.

Statistical Thermodynamics

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We know that, a proper thermodynamic system is made up or composed of an assembly of molecules or atoms, its macroscopic behavior can be explained in terms of the microscopic behavior of its constituent particles. This basic tenet provides the foundation for the subject of statistical thermodynamics. Clearly, statistical methods are mandatory as even 1 cm3 of a perfect gas contains some 1019 molecules or atoms.

Boltzmann Constant

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By definition, the Boltzmann constant is equal to the gas constant (R) divided by the Avogadro constant (NA). It is denoted by the symbol kB. The value for kB is equal to 1.380×10-23 JK-1

Application of Statistics in Physics

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It is a general belief that physics is a science where fundamental laws of nature are established. A student being introduced to physics first encounters the fundamental theories, for instance Newtonian mechanics, electrodynamics and the theory of relativity, and see in these the ideals of a scientific theory. In statistical mechanics it seems that statistics has to be taken serious. However, when one inspects theories on statistical mechanics to see how they present the concepts of probability theory. A physicist who is confronted with the interpretation of experimental results for a thesis, or who wishes to analyze the effects of complex systems in his or her later profession, will almost inevitably find that this cannot be done without the application of statistical methods. Limited information about the dynamics of a system and uncontrollable influences on the system often mean that is can only be considered as a statistical system. And apart from that, when analyzing experimental data, one always has to take into account that any measured value has an uncertainty expressed by its error. One should not only expect limited knowledge but one should also know how to deal with it. One has to be acquainted with mathematical tools allowing one to quantify uncertainties both in the knowledge about a system to be modeled as well as in the knowledge that can be inferred from experimental data of a system. These tools are probability theory and statistics. Statistics plays a special role in the description and analysis of complex systems; a physicist will need statistics not only in the context of statistical mechanics.  

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