(epistemic status: physicist, do simulations for a living)
Our long-term thermodynamic model Pn is less accurate than a simulation
I think it would be fair to say that the Boltzmann distribution and your instantiation of the system contain not more/less but _different kinds of_ information.
Your simulation (assume infinite precision for simplicity) is just one instantiation of a trajectory of your system. There’s nothing stochastic about it, it’s merely an internally-consistent static set of configurations, connected to each other by deterministic equations of motion.
The Boltzmann distribution is [the mathematical limit of] the distribution that you will be sampling from if you evolve your system, under a certain set of conditions (which are generally very good approximations to a very wide variety of physical systems). Boltzmann tells you how likely you would be to encounter a specific configuration in a run that satisfies those conditions.
I suppose you could say that the Boltzmann distribution is less *precise* in the sense that it doesn’t give you a definite Boolean answer whether a certain configuration will be visited in a given run. On the other hand a finite number of runs is necessarily less *accurate* viewed as a sampling of the system’s configurational space.
we can’t run simulations for a long time, so we have to make do with the Boltzmann distribution
...and on the third hand, usually even for a simple system like a few-atom molecule the dimensionality of the configurational space is so enormous anyway that you have to resort to some form of sampling (propagation of equations of motion is one option) in order to calculate your partition function (the normalizing factor in the Boltzmann distribution). Yes that’s right, the Boltzmann distribution is actually *terribly expensive* to compute for even relatively simple systems!
Hope these clarifications of your metaphor also help refine the chess part of your dichotomy! :)
Thanks, I didn’t know that about the partition function.
In the post I was thinking about a situation where we know the microstate to some precision, so the simulation is accurate. I realize this isn’t realistic.
(epistemic status: physicist, do simulations for a living)
I think it would be fair to say that the Boltzmann distribution and your instantiation of the system contain not more/less but _different kinds of_ information.
Your simulation (assume infinite precision for simplicity) is just one instantiation of a trajectory of your system. There’s nothing stochastic about it, it’s merely an internally-consistent static set of configurations, connected to each other by deterministic equations of motion.
The Boltzmann distribution is [the mathematical limit of] the distribution that you will be sampling from if you evolve your system, under a certain set of conditions (which are generally very good approximations to a very wide variety of physical systems). Boltzmann tells you how likely you would be to encounter a specific configuration in a run that satisfies those conditions.
I suppose you could say that the Boltzmann distribution is less *precise* in the sense that it doesn’t give you a definite Boolean answer whether a certain configuration will be visited in a given run. On the other hand a finite number of runs is necessarily less *accurate* viewed as a sampling of the system’s configurational space.
...and on the third hand, usually even for a simple system like a few-atom molecule the dimensionality of the configurational space is so enormous anyway that you have to resort to some form of sampling (propagation of equations of motion is one option) in order to calculate your partition function (the normalizing factor in the Boltzmann distribution). Yes that’s right, the Boltzmann distribution is actually *terribly expensive* to compute for even relatively simple systems!
Hope these clarifications of your metaphor also help refine the chess part of your dichotomy! :)
Thanks, I didn’t know that about the partition function.
In the post I was thinking about a situation where we know the microstate to some precision, so the simulation is accurate. I realize this isn’t realistic.