System of particles in classical mechanics and classical statistical mechanics

Question

$\bullet$ Both classical mechanics and classical statistical mechanics can describe the properties of a system of classical particles.

$\bullet$ In classical statistical mechanics, we assume that we do not know the microstate of a system at any given time. Therefore, we consider all possible microstates at any time, an ensemble.

$\bullet$ In classical mechanics, we do not consider such an ensemble i.e., a large number of copies of the same system. We consider only one system, and still get some correct results.

Below are a few questions.

1. Do you assume that we know the microstate (i.e., position coordinates and momenta of each particle) for a system of particles in classical mechanics? But since that is not true, why does the classical mechanical description work? At least it describes some properties of the system if not all.

2. Why doesn't the notion of temperature arise in the discussion of system of particles in classical mechanics?

3. Why is it that some properties of a system of particles can be described by the classical mechanical approach i.e., no ensemble? Why are some properties cannot be described by classical mechanical approach i.e. use of ensembles and microstates?

For me the holistic picture is missing. If someone could help! I am not currently asking about quantum systems just to simplify matters.

Answer 1

1) If we use some definite values of positions to find their past/future values or to find some other physical quantity, then we are doing mechanics. If we do not use any definite values of positions, then we cannot achieve the above but we can still search for probabilistic characteristics, like averages, second moments, etc. - then we are doing statistical physics. Classical mechanics description works in many situations, but not all of them. It is just a theory that helps us to understand and (in case of simple systems) predict their behaviour.

2) Which discussion? In discussion of kinetic theory, it does. Effective kinetic temperature of a system is based on average kinetic energy of the particles of that single system. However, it takes immense number of particles to make this a useful concept, so one usually uses the probabilistic description.

3) Mechanical properties of a single system such as energy, momentum, and their averages over time are mechanical concepts, so mechanics is sufficient. Probabilistic concepts such as expectated average of energy, momentum over many measurements on many different systems are probabilistic concepts, and their study requires probabilistic theory.

If you are confused about this, take it easy, first learn mechanics, then do some probability problems, then apply probability thinking to mechanics.

Answer 2

One class of systems that are of particular interest are ergodic systems. These are systems for which the time average of the phase space evolution in classical mechanics (for a fixed initial condition) is equal to the microcanonical ensemble average (an average over the energy surface in phase space) $$ \frac{1}{T} \int^T dt\, O(x_i(t),p_i(t)) = \langle O(x_i,p_i) \rangle_E $$ Basically, statistical mechanics is useful for these systems because the object on the right is easier to compute than the object on the left. Indeed, we realize that it is even easier to pass to a canonical ensemble (introduce temperature) and compute the right hand side using the equivalence between the microcanonical and the canonical ensemble. Proving ergodicity is hard, but in classical mechanics we learn about possible obstructions (integrability and invariant tori), and possible paths to ergodicity.

Answer 3

Ok to answer your first question:

in principle Classical Mechanics is capable of describing the system completely(at leas as long as we do not reach the scale of quantum world) the reason we use statistical mechanics is that when we increase the number of particles in a CM system thereby some new feature rise for instance some behavior that only depands on the interaction between the particles of our manybody system , also another difficulty is related to the chaos phenomena you can wikipedia that , and naivley speaking that occurs when we try to asymptotically use CM so these are some issues regarding to use CM that motivate people to use SM instead.

to your second question the notion of temperature arise when we talk about heat and again naively speaking heat is the energy in transfer not a specific kind of energy that most of the time it is related to the concept of friction so we use heat and therefore temperature when we do not want to specifically talking about the friction itself that one of the strength of SM that we substitute the difficult concept of friction with heat and temperature so thats why we dont talk about it in CM because there we do not seek for asymptotic answer but rather the exact one so we use friction itself .

and to your last question ensemble theory serve as a helpful tool when we cannot solve the system exactly and that is either because the equations are quite large that we do not have the computation power to do so so we use SM and ensemble theory to have some estimation of the answer the point is in principle CM is enough but because of the reasons I provide and so many others we use SM for estimation.