# What is a microstate, macrostate and thermodynamic probability in statistical mechanics?

Currently I am learning Maxwell-Boltsmann distribution (MBD) and in that I am learning about microstate, macrostate and thermodynamic probability (TDP). I understood the derivation of MBD but I am getting hard time (and also confusing) to visualise these terms (microstate, macrostate and TDP) with MBD. Can any one explain (possibly with pictures) what is microstate, macrostate and TDP and how it is used to derive MBD.

A ‘microstate’ refers to a description of the system which relies on the states of each element of the system. Applied to a thermodynamic system, each microstate $M_i$ of the system is a set of positions $\{q_i\}$ and velocities $\{\dot q_i\}$ for $i = 1,\ldots,3N$ (in three dimensions, add another set of coordinates for internal degrees of freedom, such as rotation) which describe the position and velocity of each particle. As you can imagine, for large $N$ (say, $N = 10^{23}$), this gets out of hand. Furthermore, the probability that the system is in microstate $M_i$ is quite low as there are many, many different microstates the system could occupy.

A ‘macrostate’ on the other hand is a state description relying on the macroscopic properties of the system: it’s temperature, pressure, volume, internal energy and such. For each macrostate, there are many, many microstates which result in the same macrospace: for example, if you interchange velocity (but not position) of two gas particles, the macrostate does not change, but you have a different microstate.

Concept of MICROSTATE and macro state**:for example if we have 100 balls with numbered as 1,2,3,.....100 in five boxes so that each box can have 20 balls .now box no 1 have balls from 1 to 20 and box no 2 have balls 21 to 40 .....box no five have balls from 81 to 100.now the total no of balls in each box represents macrostates i.e 20 but the individual numbered balls for corresponding a box represents the microstates.....that means A macrostate is can be assumed to be as a set of microstates.....in our example each individual numbered balls corresponding to a specified box can be treated as "microstates" each box which cotains a set of 20 no. of balls can be treated as macrostate....

Maxwell-Boltzmann distribution statistics explains what is the maximum probability of occupying the number of particles in a given macro state.Consider a system of N distinguishable particles distribution in E energy levels.N0 particles have energy of E0, N1 particles have energy of E1 .....the no. of ways N0 particles distributed in the given macro state E0 is indicated by degeneracy gi.So the thermodynamic probability (W)of distributing N no.of particles is found to be W= N!g1.g2......gN/N0!N1!.....Ni! important to note that: 1. total no of particles are constant and any no.of particles can be occupied the same energy level 2.the total energy of the system is constant.

Reference: theory.physics.manchester.ac.uk

Classical thermodynamics describes macroscopic systems in terms of a few variables (functions of state): temperature, pressure, volume... But such a system is really made of atoms, so a much richer description must be possible in principle: we could specify the quantum state of all the atoms--the microstate. Of course as the atoms interact this state changes very rapidly-perhaps 10³⁵ times a second. But the observed macrostate doesn't change. Many different microstates all correspond to the same macrostate.

This suggests we can calculate the macroscopic behaviour of the system by averaging over the corresponding microstates.

Reference: wikipedia

A microstate is a specific microscopic configuration of a thermodynamic system,that the system may occupy with certain probability in the course of its thermal fluctuation.

Macrostate: P, V, T, Density, etc.

TDP:- tdp of any particular MACROSTATE is defined as the number of MICROSTATES corresponding to that MACROSTATE. Generally it's a large no. and represented by "OHM (sign)"

• that's the multiplicity of the macrostate not the definition of the microstate itself – bhapi Dec 30 '18 at 16:49

Microstate:the number of distinct arrangements of partical in cells in phase space , Macrostate:the arrangements of partical in cells in phase space when partical are identical &the number of Microstate in a given Macrostate is called Thermodynamic probability

• You should provide more information when answering a question beyond one sentence definitions. – Brandon Enright May 8 '13 at 20:26

Macrostate- The microstate of a system is the state which can be experimentally observed. It is the state which represents the macroscopic properties of the system, and not the properties of each individual practicle of the system. Microstate- The microstate of a system is the state in which we consider the arrangement of each individual particle of the system,i.e., it is the state which represents the property of each individual practicle of the system. It cannot be experimentally observed..

Concept of MICROSTATE and macro state:for example if we have 100 balls with numbered as 1,2,3,.....100 in five boxes so that each box can have 20 balls .now box no 1 have balls from 1 to 20 and box no 2 have balls 21 to 40 .....box no five have balls from 81 to 100.now the total no of balls in each box represents macrostates i.e 20 but the individual numbered balls for corresponding a box represents the microstates.....that means A macrostate is can be assumed to be as a set of microstates.....in our example each individual numbered balls corresponding to a specified box can be treated as "microstates" each box which cotains a set of 20 no. of balls can be treated as macrostate....