# System of particles in classical mechanics and classical statistical mechanics

$$\bullet$$ 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.

• The systems that can be completely explored through classical mechanics contain a small number of particles. You cannot apply classical mechanics to a system of $10^{23}$ particles. Commented May 31, 2019 at 14:02
• @JonCuster If what you say is true, what's the purpose of the chapter "system of particles" in classical mechanics books? I also don't know what you mean by small because even a 3-body problem is also not solvable. Still we have chapter on system of particles Commented May 31, 2019 at 14:32
• It lays the foundation of classical statistical mechanics. Commented May 31, 2019 at 14:33
• 3-body problem is solvable. You just can't get an analytical expression for a solution, but e.g. JPL Horizons project has numerically calculated quite accurate ephemeris for the Solar system objects, and there's considerably more than three objects in the Solar system. Commented Jun 5, 2019 at 12:20
• Classical mechanics has no use for the concept of probability. Particles have definite and exact positions and momenta. Whether you know them or not is irrelevant. As time goes by they interact and get new exact positions and momenta. End of. Classical statistical mechanics introduces uncertainty and probability, and uses classical mechanics to describe the evolution of the ensemble. Commented Jun 9, 2019 at 23:46

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.

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.