When simulating simple system to test out Stat Mech (ie. non-coupled), I was able to compute entropy by binning the particles and using the Shannon entropy formula.

For more complex systems, I realized that entropy isn't calculated by binning the particles, but for Gibb's entropy you need to bin the state of the system over multiple simulations. A large number of particles, this is highly dimensional due to the particles and binning will be very difficult.

How is entropy typically computed molecular dynamics and similar simulations?

  • $\begingroup$ There's about a dozen different methods. A basic review can be found here, although there are plenty of other techniques. If you give more details about your system then I may be able to point you towards a particular method. $\endgroup$
    – lemon
    Oct 2, 2017 at 6:44
  • $\begingroup$ @lemon. Thanks. So my immediate problem is trying to compute entropy to verify On the classical statistical mechanics of non-Hamiltonian systems which involve integrating the Gaussian isokinetic system ( see my python notebook). Eventually, I wanted to use this in Machine Learning to constrain things like entropy and "temperature" during training. $\endgroup$
    – user92177
    Oct 2, 2017 at 16:26

1 Answer 1


If you want to get the value of entropy as given by the Gibbs-Shannon expression $\sum_k -p_k\log p_k$, you are going to have to introduce some auxiliary discrete states that the system can be in (for example, system being in a phase space cell of some small dimensions), estimate their probabilities* and then evaluate the expression.

If you are interested only on thermodynamic entropy of equilibrium systems, you can avoid that and make a simulation of quasistatic heat transfer and calculate entropy as

$$ \int_i^f \frac{dQ}{T}. $$

This gives you difference of entropy between two states, which is all that matters physically.

* probability that the system is in state $k$ given the macroscopic state variable values.


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