Deriving Statistical Behavior of Particles via Classical Mechanics

[thaiqi]

Hello, using computation simulation, can the statistical behavior of many particles be derived through deterministic classical mechanics?

 

[Nugatory]

In principle, yes. In practice... how long are you willing to wait on the computation?

Consider Boyle's law, which might be the most tractable case. Google will give you reasonable estimates for the velocity and mean free path of a particle; these will give you order-of-magnitude values for the time and space granularity you’ll need. Figure something $10^{22}$ particles in your simulation. How many floating point operations do you need to simulate one second? Divide that by what your hardware is capable of to know how long the simulation will take.

 

[Stephen Tashi]

In principle, yes.

How would we define Entropy in a deterministic simulation?

 

[Nugatory]

How would we define Entropy in a deterministic simulation?

I’m thinking the same way as always: number of microstates corresponding to a given macrostate. The simulation of course takes the system to a particular microstate, but we can still consider how many other microstates would produce the same macrostate.

 

[thaiqi]

Thanks.

In principle, yes. In practice... how long are you willing to wait on the computation?

Consider Boyle's law, which might be the most tractable case. Google will give you reasonable estimates for the velocity and mean free path of a particle; these will give you order-of-magnitude values for the time and space granularity you’ll need. Figure something $10^{22}$ particles in your simulation. How many floating point operations do you need to simulate one second? Divide that by what your hardware is capable of to know how long the simulation will take.

Can dynamical system be used to describe the behavior of the electron in the atom?

 

[dRic2]

I’m thinking the same way as always: number of microstates corresponding to a given macrostate. The simulation of course takes the system to a particular microstate, but we can still consider how many other microstates would produce the same macrostate.

Don't want to go off-topic, but you have to assume indistinguishability is not relevant to the problem you are simulating otherwise you could run into problems (like Gibbs' paradox). That's really an extreme case and is more of a question whether classical mechanics could be applied.

PS: You could still solve Gibbs paradox in the framework of classical mechanics, but you'll need to take care of the indistinguishability separately and it will not follow directly from the equations of motion (hamilton equations) since distinguishability implies a greater number of microstates. The interesting thing (to me) is that Gibbs' paradox arise in a pretty "classical" context (a box full of gas) which classical mechanics is perfectly fit to describe.

 

[vanhees71]

How would we define Entropy in a deterministic simulation?

You mean in a classical molecular-dynamics simulation? That's very difficult. One way is to consider some subsystem (e.g., considering only particles in a certain partial volume) and calculating the corresponding averages on the one-particle distribution.

 

[davenn]

Thanks.

Can dynamical system be used to describe the behavior of the electron in the atom?

So you quote one of your own threads that was locked due to your unwillingness to supply more info

What is your point ?