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Statistical Mechanics

Last updated date: 26th Feb 2024
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One of the pillars of modern physics statistical mechanics describes how microscopic observations for temperature and pressure are related parameters that fluctuate around an average. It connects thermodynamics quality which is heat and capacity to microscopic behaviour, at the same time in classical thermodynamics the only available option is to measure quantities of various materials. For the fundamental study of any physical structure that has many degrees of freedom, a statistical mechanism is needed. The approach is based on probability theory, microscopic physical law and statistical methods. Statistical mechanisms can be used to explain thermodynamic behaviour of large bodies. The branch which treated and extends classical thermodynamics of statistical mechanics is known as statistical thermodynamics or equilibrium statistical mechanics. 

Statistical Thermodynamics

Statistical thermodynamics' primary goal is to drive the classical thermodynamics of a material in terms of its properties of their constituent particles and the interaction between them. In other words we can say that the statistical thermodynamics provided a connection between the microscopic properties of materials in thermodynamic equilibrium and microscopic motion and behaviours and motion occurring inside the material. 

On the other hand statistical mechanics proper involves dynamics, here we focus on the statistical equilibrium or in steady state. Statistical equilibrium does not mean that the movement of particles is stopped, it’s rather only that the ensemble is not evolving. 

The condition for statistical equilibrium along with an isolated system is that the probability distribution is a function only of conserved properties that are total energy, number or particles. There are different equilibrium ensembles that can be considered and only some of them correspond to thermodynamics. Additional postulates are very important to motivate an ensemble for a given system should have one form or another. 

The Principle of Statistical Mechanics 

Two types of mechanics are usually examined in physics: quantum mechanics and classical mechanics. The standard mathematical approach for both the type quantum and classical two concepts are considered:

  • The first is the complete state of the mechanical system at a given time which is mathematically encoded as a phase point for classical mechanics or pure quantum state vector which is in quantum mechanics. 

  • The second one is an equation of motion that carries the state forward in time: Hamilton’s equation or the time dependent schrodinger's equation which comes under quantum mechanics. The state at any other time using these two concepts be it past or future principal can be calculated. 

This is however a disconnection between everyday experience and the law we discuss, as we do not find it necessary to know at a microscopic level this simultaneous position and velocity of each of the molecules while carrying the process at human scale. The statistical mechanics files this disconnection of the practical experience and the law of incomplete knowledge, by adding some uncertainty about which state the system is.  

Whereas on the other hand the ordinary mechanics only consider the behaviour of the single state, statistical mechanics introduces the statistical ensemble, which is a large collection of dependent, virtual copies of the system in various states. 

Foundation of Statistical Mechanics  

The pepper statistical mechanics was initiated in 1870 with the work of Boltzmann, much of which was published in his lecture on gas theory in 1896. The original paper of Boltzmann’s on statistical mechanics or interpretation of thermodynamics, the transport theory, the thermal equilibrium, H theorem, the equation of state of gases and similar objects occupy about 2000 pages in the proceeding of the Vienna academy and even other societies. The concept of equilibrium statistics was introduced by Boltzmann and also he investigated for the first time non equilibrium static mechanics. 

The term statistical mechanics was first coined by J. Willard Gibbs who was a American mathematical physicist in 1884. Probabilistic mechanism today might seem a more better and appropriate term but the statistical mechanism is firmly entrenched.  In 1902 shortly before his death, Gibbs published a book named “elementary principle in static mechanism.

Non Equilibrium Thermodynamics

This thermodynamics which is known as the non equilibrium thermodynamics is a branch of thermodynamics that deals with physics system that are not in thermodynamic equilibrium, but it can be described in terms of  variable used to specify a system in the thermodynamics equilibrium. 

The non equilibrium thermodynamics is related with the rate of chemical reaction and transport processes. The non equilibrium thermodynamics equilibrium relies on what may be thought as more or less nearness to thermodynamic equilibrium. 

All the systems almost found in nature are not in thermodynamics  equilibrium for they are triggering or changing over time and are discontinuous or continuously subject to flux of matter and energy to and from other systems to chemical reactions. 

Some processes and systems are however, in useful sence, near enough to thermodynamic equilibrium to allow description with useful accuracy by currently known non equilibrium. More general concepts are required for the study of non equilibrium systems.

FAQs on Statistical Mechanics

Q1. What's the Use of Statistical Mechanics?

Ans: The postulate is often referred to or known as the principal of equal a priori probabilities. It states that if the microstates have the same amount of the energy number of particle, volume, then they occur with equal frequency in the ensemble. 

Q2. What is the Assumption of Statistical Mechanics?

Ans: The statistical mechanism for fundamental assumption is that as time goes an isolated system in a given microstate is equally likely to be in a microstate, where energy is split up to 50-50.

Q3. State Postulate of Static Mechanics.

Ans: A Quantum insatiable is applied when the particles which are in set become indistinguishable. In classical statistics which was given by Boltzamnn, each particle has a recognizable individuality, hence the particles are distinguishable and each and every particle is likely to be in one cell or the other. 

Q4. How is Thermodynamics related to Statistical Mechanisms?

Ans: Statistical Mechanism is related with thermodynamics in such a way that it derives the oldest law and principle of thermodynamics by applying statistical methods of molecules and atoms. For a particular materials model it even allows one to determine the materials constant. 

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