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thaiqi
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Hello, using computation simulation, can the statistical behavior of many particles be derived through deterministic classical mechanics?
Nugatory said:In principle, yes.
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.Stephen Tashi said:How would we define Entropy in a deterministic simulation?
Can dynamical system be used to describe the behavior of the electron in the atom?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.
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.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.
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.Stephen Tashi said:How would we define Entropy in a deterministic simulation?
So you quote one of your own threads that was locked due to your unwillingness to supply more infothaiqi said:
"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.
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.
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.
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.
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.