Statistical Methods Applied in Physics

Statistical methods are used in statistics physics, which is a momentous interdiscipline, to provide a conceptual link between the 'macroscopic world' and the 'microscopic world'. When studying gases, we can examine the statistical distribution of particle velocities and energies by using Maxwell-Boltzmann statistics, which can be solve as a problem of combination, to gain a comprehensive understanding of the relationship between the macroscopically observable quantities and the microscopic energies of individual particles, which make up the gas. Also, applying statistical approaches to thermodynamics can lead to deeper understanding models such as Brownian motion and provide some useful insights and general ways to deal with other physical stochastic processes. Another crucial contribution of statistics to physics is Monte Carlo method, which bases on the law of large numbers and Central Limit Theorems, has applications in a wide range of fields from computational physics, molecular dynamics and related applied fields. Key Word: Brownian motion, Maxwell-Boltzmann statistics, Monte Carlo method,Ising model, Statistical Physics Introduction In recent several hundred years, the theoretical system of physics has been largely improved, part of which came from the combination of the statistics and physics, which solves problems from a brand new perspectives. And the statistical theories are applied more and more widely in Physics, have made great contributions for the development of physics. All the following content in this paper are three major contributions of statistical methods to physics. 1. Maxwell-Boltzmann statistics The main contribution of Statistics is in the statistical thermodynamics. Statistical thermodynamics is a branch of physics that applies probability theory, which contains statistics tools for dealing with large population, to probe into the thermodynamic behavior of systems consisted of a large number of particles. Statistical thermodynamics provides a conceptual link between the microscopic properties of atoms and molecules to the macroscopic materials that can be observed in everyday life. In statistical thermodynamics, two central quantities are Maxwell-Boltzmann statistics and its partition function. Maxwell-Boltzmann statistics describes the statistical distribution of particles over various energy states in thermal equilibrium, which throw a light on microstate, and Maxwell-Boltzmann statistics is valid when the temperature is high enough and density is low enough to omit quantum effects. First, suppose we have a container with plenty of particles, which have the same properties but can be distinguishable. All of these particles are moving in the container in random directions with great speed and thus possessing different energy. Therefore, the Maxwell–Boltzmann distribution is a function that explains how many particles in the container that have certain energy. To describe this system, we assume that N!(i=1,2,3...) is the occupation number of the energy level ε! (i=1,2,3...). To begin with, let’s assume that there is only one single way to put N! particles into the energy level ε!. Obviously, the number of different ways of selecting n objects from N objects without order is C!. If we have a set of boxes labeled a, b, c, d, e...z, then we select N! objects from N objects and put them in the box a (W! = C! !), and then we selects N! from the remaining N − N! objects and put them in the box b (W! = C!!!! !! ), and continuing until all objects are in boxes. The total number of the different ways is: