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What is the relation between the thermal averages in $\mu-$space and $\Gamma-$space of a system having $f$ degrees of freedom in statistical mechanics? For a system with $N$ particles (and having $n=3N$ degrees of freedom) the thermal average is given by $$ \left\langle O \right\rangle_{\Gamma} =\frac{\int O\left(q,p\right) \rho \left(q,p,t\right) \, \mathrm{d}^{n}q \, \mathrm{d}^{n}p}{\int {\rho\left(q,p,t\right) \, \mathrm{d}^{n}q \, \mathrm{d}^{n}p}} $$while the thermal average in $\mu$-space is given by $$ \left\langle O \right\rangle_{\mu} =\frac{\int O\left(q,p\right)f\left(q,p,t\right) \, \mathrm{d}^{3}q \, \mathrm{d}^{3}p}{\int f\left(q,p,t\right) \, \mathrm{d}^{3}q \, \mathrm{d}^{3}p} $$where $f\left(q,p,t\right)$ represents the single-particle phase space density (used in deriving the Boltzmann equation).

What is the relation, if any, between these two averages? Sometimes one discusses statistical properties of a system using $\mu-$space and sometimes using $\Gamma-$space which confuses me.

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    $\begingroup$ Can you explain what $\Gamma$ space and $\mu$ space are? $\endgroup$ – knzhou May 20 '18 at 20:16
  • $\begingroup$ @knzhou For a system of $N$ particles in 3-dimensional space, $\mu$ space is 6-dimensional, and the $\Gamma$-space is $6N$ dimensional. $\endgroup$ – mithusengupta123 May 21 '18 at 10:53
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    $\begingroup$ And is $O$ a function of $6$ variables or a function of $6N$ variables? $\endgroup$ – knzhou May 21 '18 at 11:20
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    $\begingroup$ I'm just not sure how these quantities are mathematically defined. Just writing "$O(q, p)"$ in both cases doesn't make sense... you might be more likely to get an answer if you just quote the part of the book you're confused about. $\endgroup$ – knzhou May 21 '18 at 13:02
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    $\begingroup$ @knzhou Actually if you look at books on Early universe cosmology they define an average in terms of single particle phase space density $f(q,p,t)$ instead of $\rho(q,p,t)$. That confuses me. I believe you agree with the first equation where $O$ is a function of $n=6N$ variables. Is that fair to assume? $\endgroup$ – mithusengupta123 May 21 '18 at 13:11
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The precise relation is given by $f(q,p,t)=\int \rho(q^N,p^N,t)\delta(q-Q)\delta(p-P)dq^Ndp^N$, where $Q$ is the center of mass of $q^N=(q_1,\ldots,q_N)$, and $P$ is the total momentum, the sum of $p^N=(p_1,\ldots,p_N)$. With this identification, both formulas give the same expectation value when applied to a 1-particle operator.

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Comment: Can you explain what Γ space and μ space are? – knzhou

Let us define $\mu$-space as phase space of one particle (atom or molecule). The macrosystem phase space ($\Gamma$-space) is equal to the sum of $\mu$-spaces.

The set of possible microstates can be presented by a continuous set of phase points. Every point can move by itself along it’s own phase orbit which lies on a surface of constant energy (ergodic surface). The overall picture of this movement possesses certain interesting features, which are best appreciated in terms of what we call a density function $\rho(q,p;t)$.

This function is defined in such a way that at any time $t$, the number of representative points in the ’volume element’ ($d^{3N}\!q \;\; d^{3N}\!p$) around the point ($q,p$) of the phase space is given by the product $\rho(q,p;t) \, d^{3N}\!q \;\; d^{3N}\!p$).

Clearly, the density function $\rho(q,p;t)$ symbolizes the manner in which the members of the ensemble are distributed over various possible microstates at various instants of time.

Question: What is the relation, if any, between these two averages? Sometimes one discusses statistical properties of a system using μ−space and sometimes using Γ−space which confuses me.

See: "Γ Space and μ Space - Two Spaces", Wikipedia's "The microstate in phase space", "Modern Thermodynamics with Statistical Mechanics, by Carl S. Helrich" (Page 155), and Wikipedia's "Thermodynamics and statistical mechanics":

"In thermodynamics and statistical mechanics contexts, the term phase space has two meanings: It is used in the same sense as in classical mechanics. If a thermodynamic system consists of N particles, then a point in the 6N-dimensional phase space describes the dynamic state of every particle in that system, as each particle is associated with three position variables and three momentum variables. In this sense, as long as the particles are distinguishable, a point in phase space is said to be a microstate of the system. (For indistinguishable particles a microstate will consist of a set of N! points, corresponding to all possible exchanges of the N particles.) N is typically on the order of Avogadro's number, thus describing the system at a microscopic level is often impractical. This leads to the use of phase space in a different sense.

The phase space can also refer to the space that is parametrized by the macroscopic states of the system, such as pressure, temperature, etc. For instance, one may view the pressure-volume diagram or entropy-temperature diagrams as describing part of this phase space. A point in this phase space is correspondingly called a macrostate. There may easily be more than one microstate with the same macrostate. For example, for a fixed temperature, the system could have many dynamic configurations at the microscopic level. When used in this sense, a phase is a region of phase space where the system in question is in, for example, the liquid phase, or solid phase, etc.

Since there are many more microstates than macrostates, the phase space in the first sense is usually a manifold of much larger dimensions than the second sense. Clearly, many more parameters are required to register every detail of the system down to the molecular or atomic scale than to simply specify, say, the temperature or the pressure of the system."

One is the average of a system and the other the average of the ensemble.

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