Deriving Statistical Behavior of Particles via Classical Mechanics

In summary, a deterministic classical mechanics simulation of a system with many particles would take a very long time to run on modern hardware.
  • #1
thaiqi
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Hello, using computation simulation, can the statistical behavior of many particles be derived through deterministic classical mechanics?
 
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  • #2
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.
 
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  • #3
Nugatory said:
In principle, yes.

How would we define Entropy in a deterministic simulation?
 
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  • #4
Stephen Tashi said:
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.
 
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  • #5
Thanks.
Nugatory said:
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?
 
  • #6
Nugatory said:
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.
 
  • #7
Stephen Tashi said:
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.
 

What is "Deriving Statistical Behavior of Particles via Classical Mechanics"?

"Deriving Statistical Behavior of Particles via Classical Mechanics" is a scientific approach that uses classical mechanics principles to explain the statistical behavior of particles, such as their motion and interactions, in a system.

How does classical mechanics explain the statistical behavior of particles?

Classical mechanics uses mathematical equations, such as Newton's laws of motion, to describe the behavior of individual particles in a system. By applying these equations to a large number of particles, the statistical behavior of the system as a whole can be derived.

What are some limitations of using classical mechanics to study particle behavior?

One limitation is that classical mechanics cannot fully explain the behavior of particles at the quantum level. It also assumes that particles follow deterministic paths, which may not always be the case in reality.

How is "Deriving Statistical Behavior of Particles via Classical Mechanics" relevant to other fields of science?

This approach has applications in various fields, such as thermodynamics, fluid dynamics, and astrophysics. It can also be used to study complex systems, such as gases and fluids, and make predictions about their behavior.

What are some current research areas related to "Deriving Statistical Behavior of Particles via Classical Mechanics"?

Some current research areas include using this approach to study non-equilibrium systems, developing new mathematical techniques to improve the accuracy of predictions, and applying it to study biological systems.

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