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I'm currently studying statistical mechanics and having a hard time going conceptually from the canonical ensemble to the gran canonical ensemble. Up until now I've been studying only ensembles with a fixed number of particles therefore I've been thinking about the system's state as living in a 6N-dimentional phase space. Moreover the probability density function for the ensemble was a function on phase space.

Where does the state of a system with a varying number of particles live in? What's the equivalent of phase space for the gran canonical ensemble? How is the ergodic hypothesis formulated in this new space?

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    $\begingroup$ Since the number of particles can be any whole number, there is no single usual $6N$-dimensional phase space. Instead, the grand canonical ensemble refers to a probability distribution describing all possibilities $N=0,N=1,...$, thus one can say all possible phase spaces for all possible sets of particles are considered. The resulting "superspace" of states the control volume can be in is kind of similar to what the Fock space is in quantum theory - possible states describe all possible numbers of particles in all possible single-particle states. $\endgroup$ Commented Dec 29, 2023 at 13:15
  • $\begingroup$ So basically you work on the union of all possible phase spaces as N goes from $0$ to $+\infty$. A measure for this set can be easily defined so I guess integrating the probability density function would not be a problem. Tough, how to generalize the ergodic hypothesis is still unclear... $\endgroup$
    – HomoVafer
    Commented Dec 29, 2023 at 13:22
  • $\begingroup$ Why would you want to "generalize the ergodic hypothesis"? It originated in mathematics of isolated systems. Already canonical ensemble describes a system interacting with a heat reservoir. Ergodicity seems immaterial to this description. $\endgroup$ Commented Dec 29, 2023 at 13:51
  • $\begingroup$ About the ergodic hypothesis I think you are right: the question doesn't make sense since the hypothesis is specif for micro-canonical (isolated) ensembles. Thanks! $\endgroup$
    – HomoVafer
    Commented Dec 29, 2023 at 15:17

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The correct phase space $X$ of a classical system describing an arbitrary number of identical particles is $$ X = \coprod_{n \in \mathbb{N}} \mathbb{R}^{6n}, $$ where $\coprod$ denotes the disjoint union. An element of $X$ has the form $(n,x)$ where $n\in \mathbb{N}$ is the number of particles and $x\in \mathbb{R}^{6n}$ is the vector of momenta/positions of the $n$ particles.

Now there is not a single probability density function. Instead there is a function $\rho_n : \mathbb{R}^{6n} \to [0,\infty)$ for every $n \in \mathbb{N}$ such that the probability $P(n,\Gamma)$ to observe the system consisting of $n$ particles with their momenta and positions in the phase space volume $\Gamma \subset \mathbb{R}^{6n}$ is $$ P(n,\Gamma) = \int_\Gamma \rho_n (x) dx^{6n}. $$

Note that the normalisation condition here is, that $$ 1 = \sum_{n=0}^\infty \int_{\mathbb{R}^{6n}} \rho_n (x) dx^{6n}, $$ i.e. the probability of observing any number of particles with any position and momenta is 1.

I am not sure about the "ergodic hypothesis" in this context, maybe you can elaborate what exactly the "ergodic hypothesis" means to you.


Elaborating on the ideas discussed in the comments (in particular HTNW's comment):

In the case of multiple kind (species) of particles the phase space can be constructed as follows:

Let $S$ be the (finite) set of species. Define the state space $X$ of a classical system describing an arbitrary number of particles from different species (from $S$) as $$ X = \prod_{s \in S} \coprod_{n \in \mathbb{N}} \mathbb{R}^{6n}. $$ So an element of $X$ (a state of the system) is a tuple of the form $(n_s,x_s)_{s \in S}$, where $n_s$ is the number of particles of species $s$ in the state and $x_s$ is the vector storing positions and momenta of these $n_s$ particles.

A probability measure can be constructed on this space by considering the product of the probability measures defined above for each species.

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  • $\begingroup$ Do you mean indistinguishable or distinguishable particles? For indistinguishable particles, one has to impose further restrictions on $\rho_n$. For $n$ distinguishable particles, one should consider all possible phase spaces $\mathbb{R}^{6n}$, there is not just one. $\endgroup$ Commented Dec 29, 2023 at 13:48
  • $\begingroup$ @JánLalinský The particles are supposed to be of the same species. Different species can be incorporated by taking an additional disjoint union over all present species. $\endgroup$
    – jd27
    Commented Dec 29, 2023 at 14:53
  • $\begingroup$ @JánLalinský That being said i do not understand your comment and how it relates to my answer/ the grand canoncial ensemble of a classical system at all, please elaborate. $\endgroup$
    – jd27
    Commented Dec 29, 2023 at 14:59
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    $\begingroup$ You use $\mathbb R^{6n}$ to refer to a phase space of $n$ particles, but there are many different spaces of such dimension, for different sets of particles. $\endgroup$ Commented Dec 29, 2023 at 15:09
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    $\begingroup$ @jd27 You would take a product over the species and a sum over the numbers, no? The state space for a system with varying particle numbers where each particle has a species (from a set $S$ of species) will be $X=\prod_{s\in S}\coprod_{n\in\mathbb N}\mathbb R^{6n}.$ A state will be a vector $(n_1,x_1,\ldots,n_s,x_s)$ where $n_i$ is the number of particles of the species indexed with $i$ and $x_i$ are the coordinates of those particles. Alternatively, the state is a function from the set of species to the set described in the answer. $\endgroup$
    – HTNW
    Commented Dec 29, 2023 at 20:26

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