How to calculate entropy from positions and velocities of gas molcules

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Discussion Overview

The discussion centers on how to calculate entropy from the positions and velocities of gas molecules, particularly in the context of mixing two different gases. Participants explore the definitions of microstates and macrostates and their relationship to entropy calculation.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how to calculate exact entropies given only the positions and velocities of gas molecules.
  • Another participant asserts that entropy is defined as the log of the number of microstates for a given macrostate, indicating that the macrostate must be defined to calculate entropy.
  • A participant challenges the notion that macrostates are determined solely by microstates, suggesting that knowing positions and velocities should provide complete information about the gas's state.
  • Further clarification is sought regarding the relationship between macrostates and microstates in the context of entropy calculation.
  • One participant proposes that if the microstate is close to thermal equilibrium, it might be possible to determine pressure and temperature, which could then allow for entropy calculation, although this is not the standard approach.
  • Another participant emphasizes that entropy is typically defined using macroscopic variables, and that if the microstate exhibits rapid variations, entropy may not be meaningfully defined.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the relationship between microstates and macrostates or on how to calculate entropy from the given information. Multiple competing views remain regarding the definitions and implications of these concepts.

Contextual Notes

There are unresolved questions about the definitions of macrostates and microstates, as well as the conditions under which entropy can be calculated meaningfully. The discussion highlights the complexity of relating microscopic properties to macroscopic thermodynamic quantities.

olgerm
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How to calculate entropy from positions and velocities of gas molecules?

lets say we have 2 different gases. entropy should be bigger after mixing them, than before when these are separated. But how to calculate exact entropies by knowing only positions and velocities of gas molecules?
 
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This is not defined.

Entropy is the log of the number of microstates for a given macrostate. You have defined the microstate but not the macrostate.
 
Vanadium 50 said:
You have defined the microstate but not the macrostate.
isnt macrostate determined by the microstate? By knowing positions and velocities of gas molecules, I should know everything about the state of gas(including marcostate). Or you mean I should define possible(measurable) macrostates and microstates that correspond to every macrostate?
 
olgerm said:
isnt macrostate determined by the microstate?

Of course not.

Suppose you had two otherwise identical particles, one moving to the left and one to the right at 1 m/s. Is your macrostate:
  • All states of two particles?
  • All states with an even number of particles?
  • All states with zero net momentum?
  • All two particle states with velocity differences of 2 m/s?
  • All two particle states with x-component of velocity differences of 2 m/s?
  • All states with p = 0 and E = m?
 
Vanadium 50 said:
Is your macrostate:
I don't know.

Do you mean that I need relation between macrostates and mircrostates or macrostate of given microstate to calculate entropy?
 
Last edited:
olgerm said:
Do you mean that I need relation between macrostates and mircrostates or macrostate of given microstate to calculate entropy?

Vanadium 50 said:
Entropy is the log of the number of microstates for a given macrostate.

So yes, you need to know your macrostate. You have not defined the problem.
 
In principle, if the microstate is "close enough to thermal equlibrium", with the particle density and kinetic energy density being constant down to very small scale, it could be possible to determine the pressure and temperature corresponding to that microstate and then calculate the entropy. But that's not how it's usually done and does not reflect the actual meaning of the concept of entropy. Entropy is a function of the macroscopic variables like volume, pressure and temperature, and can be defined without any knowledge about the atomic nature of matter.

If the microstate is such that there are rapid variations in density and the velocity distribution of the molecules isn't anywhere near the Boltzmann distribution, variables like entropy and temperature can't be defined in a meaningful way.
 
Last edited:

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