Question

My textbook (Moore Physical Chemistry) defines a microcanonical ensemble by considering an isolated system containing N molecules of gas with fixed volume V and energy E. It states we can construct an ensemble of N

such systems, each member having the same N, V, E and this is called a microcanonical ensemble. It then states two postulates for such an ensemble:

The time average of any mechanical variable M over a long time in the actual system is equal to the ensemble average of M (the book notes that strictly speaking, this only holds in the limit N→[infinity])

The members are distributed with equal probability over the possible quantum states consistent with the specified values of N, V, E.
I have a few questions about this definition and the following postulates.

It is not explicitly stated how many members N the ensemble must contain. The book states that we can have any number of members of the ensemble to represent each state of the system (so long as it is the same number for each one). I take this to mean that the ensemble must contain at least one member for each possible quantum state the system may attain (the word state seems to mean many different things (e.g. state of a system, state of an individual particle, microstate, macrostate, etc.) so I am not fully sure what it is referring to in the previous quote). However, I am not sure what the significance of having multiple members of the ensemble with the same quantum state as it seems redundant (in the sense that having multiple copies of the same quantum state in the ensemble will not contribute anything to the ensemble average because the states are exactly the same). Furthermore, to consider the limit N→[infinity] required for the first posulate, it seems that the ensemble under consideration must contain infinitely many systems corresponding to each of the possible quantum states and again I don't see how the ensemble average of properties for this ensemble will be any different from the ensemble average of properties from the ensemble containing exactly one system for each possible quantum state.

When considering microcanonical ensembles, is the word microstate synonymous with one of the systems in the ensemble or is there a difference.

I am not fully sure how to rationalize the idea of a microcanonical ensemble within the realm of classical mechanics. In classical mechanics, the system can (in theory) occupy an uncountably infinite number states with the same N, V, E
. The book describes: in classical mechanics, we can imagine phase space to be divided into cells of volume δτ; we can then imagine a microscopic state to be specified by giving the particular cell in phase space in which the system is situated. I was hoping someone could explain what exactly is meant by this phrase as I would imagine a microstate to involve a collection of points in phase space, representing the individual states of each of the particles, not an element of a discrete volume cell. Furthermore, it is not immediately clear why each of these cells must correspond to systems with the same N, V, E
.

Answer 1

A microcanonical ensemble is a large collection of identical systems representing all possible microstates with equal probability. The concept helps define macroscopic thermodynamic quantities statistically. Ludwig Boltzmann's formula relates entropy to the number of microstates by S = k ln W.

Step-by-step explanation:

The concept of a microcanonical ensemble originates from statistical mechanics, where it is used to represent the statistical behavior of an isolated system with fixed number of particles (N), volume (V), and energy (E). This ensemble is composed of a large number of hypothetical copies, or members, of the system, each representing one possible microstate of the system. A microstate is a specification of all the microscopic details of a system, including the positions and momenta of all particles.

The significance of having multiple members of the ensemble in identical microstates is not to provide variety in the ensemble average but rather to indicate the probability associated with each microstate. If microstates have equal probability, as in a microcanonical ensemble, the number of members in each microstate reflects this equiprobability. When we consider the limit as the number of particles N approaches infinity, we are exploring the behavior of macroscopic systems and applying the laws of thermodynamics in a statistical sense.

In the realm of classical mechanics, the phase space of a system is continuous and can be broken down into discrete cells, each corresponding to a unique microstate. If these cells are sufficiently small (τ), they will effectively represent the continuous range of possible microstates, since each tiny volume contains a distinct configuration of positions and momenta. Idea of dividing phase space into cells helps to discretize an otherwise continuous phase space, making it possible to apply statistical methods similar to those used in quantum mechanics.

In summary, a microcanonical ensemble is a collection of identical systems representing all possible microstates with equal probability, allowing us to define macroscopic thermodynamic quantities, such as entropy, in terms of microscopic configurations. Ludwig Boltzmann's formula S = k ln W, where S is entropy, k is the Boltzmann constant, and W is the number of microstates, quantifies this relationship.