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Pathria, Statistical mechanics, 4ed,pg32-33

"The microstate of a given classical system, at any time, may be defined by specifying the instantaneous positions and momenta of all the particles constituting the system. Thus. If $N$ is the number of particles in the system, the definition of a microstate requires the specification of $3 N$ position cuordinates and $3 N$ momentum coordinates. Geometrically, the set of coordinates $(q, p)$ may be regarded as a point in a $6N$ dimensional phase space"

( $\omega$ be a "volume" in the phase space ) the author says

"The next thing we look for is the establishment of a connection between the mechanics of the microcanonical ensemble and the thermodynamics of the member systems. To do this, we observe that there exists a direct correspondence between the various microstates of the given system and the various locations in the phase space. The volume $\omega$ (of the allowed region of the phase space) is, therefore, a direct measure of the multiplicity $\Gamma$ of the microstates accessible to the system. To establish a numerical correspondence between $\omega$ and $\Gamma$, we need to discover a fundamental volume $\omega_{0}$ that could be regarded as "equivalent to one microstate. Once this is done, we may say that, asymptotically, $$ \Gamma=\omega / \omega_{0}" $$

($ \Gamma$ is the number of microstates calculated quantum mechanically.)

Since for one microstate we have one point in phase space then how does there exist a volume in phase space that corresponds to one microstate?

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  • $\begingroup$ In classical physics the positions and momenta can be continuously variable, so a state of the system corresponds to a point in phase space. In quantum mechanics the allowable values of momentum may be quantised, so there are empty spaces in the phase space which correspond to momentum values that are not allowed. Each allowed state is still a point, but if you take into account the average amount of empty space around it, you can allocated a volume to each space. $\endgroup$ Apr 9, 2022 at 11:03

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The author might mean that he wants to change the discussion of purely classical statistical physics (where states are represented as points in continuous phase space) into pseudo-quantum statistical physics, where the state is represented as point in discrete space of states. Which makes lots of calculations easier, and some calculations have very different results.

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  • $\begingroup$ So in the pseudo qm formulation we've the 6N dimensional space chopped into small discrete cubes each representation of a micro state? $\endgroup$
    – Kashmiri
    Apr 9, 2022 at 5:28
  • $\begingroup$ I've added more relevant information, the author does indeed bring in qm. Please see the edit $\endgroup$
    – Kashmiri
    Apr 9, 2022 at 6:07
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In classical physics momentum and position can be measured simultaneously with full accuracy so there is no limit on the volume of the lowest possible phase space cell , it can be as smaller as a point . But in quantum physics the volume of the lowest possible phase space cell cannot be less than planks constant because of uncertainty principle

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