# How can I compute the longest relaxation time?

Cross-posted on Stats.SE and on MMSE.

In the case of Monte Carlo simulations:

• Autocorrelation Time ($$\tau_{\text{int}}$$): A measure of how many steps are needed for the correlations in the chain to become small; it affects the variance of estimators based on the Markov chain.

• Relaxation Time ($$\tau_{\text{relax}}$$): The time it takes for the Markov chain to approach its equilibrium distribution from an arbitrary starting point.

• Longest Relaxation Time ($$\tau_1$$): The slowest timescale on which the system relaxes to equilibrium, which is critical for understanding the overall convergence rate of the Markov chain.

I realize that autocorrelation time is the sum of autocorrelation values over a given maximum lag time. So, the formula becomes:

$$\tau_{\text{int}} = \frac{1}{2} + \sum_{t=1}^{\text{max-lag}} ACF(t)$$

where $$ACF()$$ is the autocorrelation function:

$$ACF(k) = \frac{1}{(n-k)\sigma^2} \sum_{i=1}^{n-k} (x_i - \mu)(x_{i+k} - \mu)$$

I also realize that, relaxation time can be found by fitting an autocorrelation value to the exponentia decay function ($$A_0 \cdot e^{\frac{-t}{\tau_\text{relax}}}$$).

How can I compute the longest relaxation time?

• Please put a banner at the top of your question that has a link to all cross-postings, not just the MMSE one but also the Stats one. Mar 2 at 12:17
• Is the longest relaxation time actually well defined? It was my understanding that Monte Carlo Methods will never converge to a single defined value due to the randomness of the samples. Isn't the longest relaxation time simply the maximum time (or samples) which were considered in the Monte Carlo Simulation? Mar 4 at 14:50
• @MPIchael, you tell me. Mar 4 at 16:56