# 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.

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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.

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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)"

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