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I have been puzzled with the definition of ensemble in Statistical Mechanics. Different sources define it in different ways, e.g.,

  • Introduction To Statistical Physics (Huang), Thermodynamics and Statisitical Mechanics (Greiner): ensemble is a set of identical copies of a system (characterized by some macroscopic variables), each of which being one of the possible microstates of the system.

  • Introduction to Modern Statistical Mechanics (Chandler): ensemble is a set of all possible microstates of a given system, consistent with the macroscopic variables of the system.

  • Statistical Mechanics in a Nutshell (Peliti), Statistical Physics (lectures by D. Tong): ensemble is a probability distribution.

  • Wikipedia: at the beginning of the article, ensemble is also defined as a collection of identical copies of a system; afterwards, it is said to be a probability space.

It seems to me that the correct definition is that of a probability space. I tried to translate the mathematical definition of probability space in more intuitive terms: it is a triple composed by: a sample space $\Omega$ (set of all possible outcomes of a experiment, or microstates) , event space (set of all subsets $\Omega$, it subset being a macrostate) and a probability law (a function that assigns a number between o and 1 to a element of the event space), and satisfies the Kolmogorov axioms.

My questions are, please:

1) What is the correct definition of an ensemble?

2) Should indeed ensemble=probability space be the correct definition, is my "translation" of ensemble=probability space correct? In particular, I am not sure about the interpretation of a element of the event space as a macrostate.

3) How does the concept "identical copies" appears if one considers the definition of ensemble=probability space?

4) Does anyone knows a less sloppy reference regarding the definition of ensemble?

Regards!

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    $\begingroup$ The phase space of a dynamical system is a manifold. Formally speaking, an ensemble is then a measure on this manifold. If you now imagine an infinity of copies of this system, possibly an uncountable infinity, each represented by a point in the phase space, you can interpret the measure of a certain region of the phase space as the fraction of copies of the system whose state lies in that region. $\endgroup$ Commented Aug 30, 2016 at 18:47
  • $\begingroup$ Just to add to the comment by MassimoOrtolano, we can have a specific 'microstate' for (E,V,N), such that prior to de-labelling (we know the specifics for each particle) there is a unique configuration. There are multiples of these configurations 'microstates' which satisfy the same (E,N,V), so that we have a set of them. The ensemble is the set of all these microstates that satisfy the macrovariable observation (E,N,V) that does not look at other specifics of the configurations. This expands upon your second bullet point, and the manifold allows us to do integration on the space $\endgroup$
    – Vass
    Commented Jun 26, 2018 at 14:07

2 Answers 2

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I believe it is easier to get the general idea if we first contemplate the problem it solves.

  • Let's begin by saying we're interested in tracking the trajectories of a multitude of particles and see if we're able to tell something about them. We will need time averages, since trajectories are functions of time;
  • We notice our particle number is so large that we're pretty much unable to deal with anything that involves trajectories, initial conditions, etc;
  • We now focus on the fact that our system follows the laws of mechanics and there are some interesting theorems that might help us a lot, like Poincaré's recurrence theorem and the conservation of energy;
  • Keeping recurrence in mind we postulate, on physical grounds, that the multitude of orbits we couldn't track before are now so incredibly complicated that their complexity happens to help: we say they will recur and that each and every region accessible to them will be filled;
  • Since each and every region is, according to our assumption, accessible, we no longer need time averages. We can use space averages instead. This clearly emphasizes need for the notion of some space where each point is a possible configuration of our system in phase space, that is, a probability space formed over our initial phase space;

I will now define such a probability space in a manner that looks proper to me. We will start defining $M$ as the even dimensional manifold of our system, which mathematically is a symplectic dynamical system $D=(M,\omega,T_n)$, where $\omega$ is the symplectic 2-form and $T_t$ is our dynamics, that is, the law which dictates the behaviour of each of our particles as a function of time. Since this law was rendered useless I'm not at all concerned with it, but we must remember that we tacitly assumed this law was weakly mixing or at least ergodic when we substituted time averages for space averages... Fortunately, for Hamiltonians systems this is true: all the energy surface will be densely filled with trajectories.

Now, let us take our symplectic dynamical system $D$ and imagine all possible configurations it might access, as described before. We do this by creating a power set $C(M)$ of all possible phase space states we can find, and I claim that

  1. If one configuration is present in this set, then all other configurations complementary to it are, too (of course, because we assume all configurations are possible);
  2. The countable union of configurations is still a configuration, since they are all allowed (I implicitly used the fact that we are considering an infinite number of particles here);

As the last step, since I'm interested in integration, I do some analysis and notice that providing a Lebesgue measure $\mu$ to this space makes sense. We have thus created the ensemble $(M,C(M),\mu)$, which was built upon the notion of a measure space, $M$ being the topological space, $C(M)$ a $\sigma$-algebra and $\mu$ a (finite) Lebesgue measure over it, which can be turned into a probability measure.

I emphasize this is not rigorous. There are flaws and bypasses I took a mathematician would call "cheating", but I'm not a mathematician. Thinking on these terms has helped me a lot to understand the foundations of Statistical Mechanics. Also, you didn't find this clearly exposed nowhere else because physicist usually don't care about and mathematicians usually don't use it: they prefer using a formalism that applies central limit theorem instead (where everything is indeed much clearer). For a glimpse, check Khintchin's book.

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  • $\begingroup$ Has rendered my English useless. hahah It was a typo. Just corrected it. I'll still change some things in this answer. $\endgroup$ Commented Aug 30, 2016 at 20:32
  • $\begingroup$ @QuantumBrick Thanks for the reference, it looks indeed a nice and deep book! Unfortunately, I cannot fully understand your answer, since I am not familiar with symplectic Mechanics. But, if I got correctly, you defined ensemble as some sort of probability space (your $(M,C(M),\mu)$), is that correct? If yes, what would be a macroscopic state (if it is the case) in this context? $\endgroup$
    – aprendiz
    Commented Aug 30, 2016 at 21:03
  • $\begingroup$ Yes, I tried to motivate the definition of a $\sigma$-algebra of states, which is where probability happens. Each state is macroscopic but composed by microscopic particles, such that all possibilities of position and momentum are covered in $C(M)$. The context is the same you're used to, I only tried to show that this ensemble of possible states looks (and is) a probability space. $\endgroup$ Commented Aug 30, 2016 at 21:07
  • $\begingroup$ @QuantumBrick can you give me a reference about ergodic and Hamiltonian flows? I'm seeing the ergodic hypothesis everywhere, but I'm in need of some justification on when can this be assumed, and I need to know when Birkhof's ergodic theorem can be applied. $\endgroup$
    – nabla
    Commented Feb 6, 2017 at 13:07
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Ensemble is a collection of objects.The particular notion of statistical ensemble was coined first for the study of Equilibrium Statistical Mechanics ESM in gaseous systems. Practically a gaseous system is composed of very large number of molecules.Obviously it is impossible to solve such indefinitely high number of classical Newtonian mechanics equations. Therefore, In order to study such systems :

Boltzmann and Gibbs firstly introduced the notion of statistical ensemble which is a collection of all possible micro-states corresponding to a well defined macroscopic state,i.e macro-state defined by its total energy E,number of molecules N and volume V. In agreement with the second definition which is the exact one among the four proposed by the question.

2-They assumed that the Ergodic Theory stating that the time average of the system exists uniquely and coincide with the ensemble phase space average.

3- In accord with the principle of equal a prior probability they defined the probability of a system in a micro-state as proportional to the number of its identical copies or replicas.

4-They postulated that an individual system spends its time intervals among different micro-states in proportionality to the probability of each of them.

The postulates 1–4 form the basis of the Equilibrium Statistical Mechanics ESM where we should"with utmost probability near to 1", find the system in its most probable micro-state of its canonical or micro-canonical ensemble. As an answer to part 3 of the question consider a simple system of 4 molecules in a micro-canonical ensemble having total energy of four units and the energy levels of molecules are discrete 1u,2u,3u,4u.A possible micro-state is (0,4,0,0),another on is (2,0,2,0).It is obvious that the first micro-state has only one copy while the second has 6 .

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