Reweighting Simulations for Temperature Extrapolation

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to the next lesson in our module on computational methods.

In our last session, we equipped ourselves with the Wolff algorithm, a powerful tool to overcome critical slowing down and efficiently generate statistically independent configurations of a system near its critical point. Having a good data generator is the first half of the battle. The second half is extracting the maximum possible information from that data. Running large-scale simulations is computationally expensive, and it would be a tremendous waste of resources to perform separate long runs for every single temperature point we wish to investigate.

Today, we address this challenge head-on. Our learning outcome is to apply the histogram reweighting technique to extrapolate simulation results to nearby temperatures from a single run. This powerful analysis method, pioneered by Ferrenberg and Swendsen, allows us to use the data from a single simulation at one temperature to calculate thermodynamic properties over a continuous range of nearby temperatures, dramatically increasing the efficiency of our numerical studies.

1. The Fundamental Principle of Reweighting

Imagine you have performed a long Monte Carlo simulation at a fixed inverse temperature $β_{0} = 1/ (k_{B} T_{0})$ . This simulation has generated a sequence of configurations ${ϕ_{i}}$ , which are representative samples from the canonical Boltzmann distribution at that temperature:

p_{β_{0}} (ϕ) = \frac{e ^{- β_{0} E (ϕ)}}{Z ( β _{0} )}

where $E (ϕ)$ is the energy of configuration $ϕ$ and $Z (β_{0})$ is the partition function.

The question is: can we use this set of configurations, sampled at $β_{0}$ , to calculate the expectation value of an observable $O$ at a different inverse temperature, $β^{'}$ ? The answer is yes, and the derivation is surprisingly direct.

To see how, let's start by exploring the foundational equations of this technique.

Reweighting

This resource provides a concise and clear derivation of the core reweighting formula. It shows how an expectation value at a new temperature can be expressed in terms of expectation values at the original simulation temperature.

Please read the first section, '6 Reweighting', up to the final equation. Focus on how the probability distribution p_β'(φ) is related to p_β(φ) and follow the derivation to obtain the final expression for ⟨O⟩_β'.

As the reading shows, the expectation value of an observable $O$ at the new temperature $β^{'}$ can be expressed entirely in terms of expectation values calculated at the original simulation temperature $β_{0}$ . The final result is:

⟨ O ⟩_{β^{'}} = \frac{⟨ O e ^{- (β^{'} - β_{0}) E} ⟩ _{β_{0}}}{⟨ e ^{- (β^{'} - β_{0}) E} ⟩ _{β_{0}}}

This is the central theoretical statement of the reweighting method. It tells us that if we measure the energy $E$ of each configuration, we can "re-weight" its contribution to the thermal average to determine what its contribution would have been at the new temperature $β^{'}$ .

2. From Theory to Practice: Applying Reweighting to Simulation Data

The formula above is an exact relation between ensemble averages. In a simulation, we don't have the true ensemble average; we have a finite sample of $N$ measurements drawn from the distribution at $β_{0}$ . We can use this sample to estimate the expectation values.

This leads to a practical and powerful recipe for analyzing Monte Carlo data.

Monte Carlo Simulations in Statistical Physics

The following text translates the theoretical formula into a practical algorithm for use with Monte Carlo simulation data. It also provides a more general perspective on how to handle observables that are not simple functions of energy.

Please read section 4.1.2, 'Reweighting of non-conjugate observables'. Focus on the time-series method described by equations (72) and (73). This is the most general and widely used approach in practice.

As the resource explains, the most flexible approach is to store the time series of your measurements. If you have run a simulation at $β_{0}$ and recorded a series of $N$ measurements of the energy ${E_{1}, E_{2}, \dots, E_{N}}$ and an observable ${O_{1}, O_{2}, \dots, O_{N}}$ , you can approximate the ensemble averages in our reweighting formula with simple averages over your data:

⟨ O ⟩_{β^{'}} \approx \frac{\sum _{i = 1}^{N} O _{i} e ^{- (β^{'} - β_{0}) E_{i}}}{\sum _{i = 1}^{N} e ^{- (β^{'} - β_{0}) E_{i}}}

This is the workhorse formula for histogram reweighting. The practical workflow is as follows:

Simulate: Perform a single, long Monte Carlo run at a carefully chosen inverse temperature $β_{0}$ (often chosen to be near the estimated critical temperature). Use an efficient algorithm like Wolff to ensure your samples are decorrelated.
Record: During the simulation, after equilibration, write the measured energy $E_{i}$ and the value of any observable of interest $O_{i}$ (e.g., magnetization $M_{i}$ ) to a file for each configuration $i$ you sample.
Analyze: After the simulation is complete, write a separate analysis script. This script reads the file of ${E_{i}, O_{i}}$ pairs. It then loops over a range of desired $β^{'}$ values around $β_{0}$ and uses the formula above to compute $⟨ O ⟩_{β^{'}}$ for each $β^{'}$ .

The result is a nearly continuous curve of $⟨ O ⟩$ as a function of temperature, all generated from one initial simulation. You can use this to calculate the specific heat $C_{V} (T)$ or susceptibility $χ (T)$ and find the precise location of their peaks without needing to guess the peak location beforehand.

3. An Intuitive View: Reweighting Histograms

The name "histogram reweighting" comes from an alternative, more intuitive way to view the process. During a simulation at $β_{0}$ , we can build a histogram of the visited energy levels, which is an estimate of the energy probability distribution $P_{β_{0}} (E)$ . We know that this distribution is a product of two factors:

P_{β_{0}} (E) \propto Ω (E) e^{- β_{0} E}

Here, $Ω (E)$ is the density of states—the number of microscopic configurations that have energy $E$ . This quantity is a fundamental property of the Hamiltonian and is independent of temperature. The exponential term is the familiar Boltzmann factor.

A simulation at $β_{0}$ provides a window into the function $Ω (E)$ over the range of energies that are probable at that temperature. By inverting the relation, we can get an estimate for the density of states: $Ω (E) \propto P_{β_{0}} (E) e^{+ β_{0} E}$ . Once we have an estimate for $Ω (E)$ , we can predict the energy histogram at any other temperature $β^{'}$ :

P_{β^{'}} (E) \propto Ω (E) e^{- β^{'} E} \propto (P_{β_{0}} (E) e^{β_{0} E}) e^{- β^{'} E} = P_{β_{0}} (E) e^{- (β^{'} - β_{0}) E}

This shows that we are literally "re-weighting" the bins of the energy histogram measured at $β_{0}$ to get the histogram for $β^{'}$ .

Histogram Reweighting Approach Illustration — This diagram illustrates the underlying principle. A simulation samples a range of energies determined by the Boltzmann factor at its temperature. The histogram of visited energies gives us a view of the underlying density of states $\Omega(E)$ (gray curve). By combining data from simulations at different temperatures ($\beta_1, \beta_2, \beta_3$), we can piece together a more complete picture of $\Omega(E)$. Single-histogram reweighting uses the data from just one of these runs to extrapolate into the regions covered by its neighbors.

This histogram-based view is excellent for intuition, but as noted in the resources, it is less general than the time-series method, which easily handles observables (like magnetization) that are not solely functions of energy.

4. Practical Limits: How Far Can We Reweight?

Reweighting seems almost magical, but it has important limitations. The technique relies on the assumption that the configurations sampled at $β_{0}$ are also reasonably probable at $β^{'}$ . If $β^{'}$ is too far from $β_{0}$ , the energy range important for the new temperature will lie in the far tails of the original energy distribution, where you have poor statistics or no data at all.

Reweighting

Understanding the limits of reweighting is crucial for its correct application. This section provides a practical rule of thumb and derives how the valid reweighting range scales with the system size.

Please read section 6.3, 'How far one can reweight?'. Pay close attention to the argument that the shift in the average energy must be smaller than the fluctuation of energy in the original simulation. Follow the derivation of how the allowed range \Deltaeta scales with volume V both at and away from a critical point.

The key insight is that the reweighting is reliable only as long as the reweighted energy distribution has significant overlap with the original one. A good rule of thumb is that the shift in the average energy at the new temperature, $∣ ⟨ E ⟩_{β^{'}} - ⟨ E ⟩_{β_{0}} ∣$ , should not be much larger than the standard deviation of the energy in the original simulation, $Δ E_{0} = ⟨ E^{2} ⟩_{β_{0}} - ⟨ E ⟩_{β_{0}}^{2}$ .

Using the relation $C_{V} = (β^{2} / V) (⟨ E^{2} ⟩ - ⟨ E ⟩^{2})$ and a first-order Taylor expansion for $⟨ E ⟩_{β^{'}}$ , this condition leads to an estimate for the valid reweighting range $δ β = ∣ β^{'} - β_{0} ∣$ :

δ β ≲ \frac{1}{V C _{V}}

This has important consequences for how the technique scales with system size $V = L^{d}$ :

Off-critical: $C_{V}$ is finite, so the reweighting range shrinks as $V^{- 1/2}$ .
At a critical point: $C_{V}$ diverges with system size, $C_{V} \sim L^{α / ν}$ . The range $δ β$ shrinks even faster. This means that for larger systems, you need to perform simulations at temperatures closer together.

Test your understanding!

You perform a large simulation of the 3D Ising model on a $6 4^{3}$ lattice at $β_{0}$ , very close to the critical point $β_{c}$ . You use reweighting to calculate the magnetic susceptibility $χ (β)$ and find its peak. A colleague runs a simulation on a much smaller $1 6^{3}$ lattice at the same $β_{0}$ . Will your valid reweighting range be larger, smaller, or about the same as your colleague's? Why?

Show answer

Your valid reweighting range will be significantly smaller. The formula $δ β ≲ 1/ V C_{V}$ shows that the range depends on both the volume $V$ and the specific heat $C_{V}$ . Near criticality, $C_{V}$ itself grows with volume. Since your volume $V$ is much larger ( $6 4^{3}$ vs $1 6^{3}$ ), and $C_{V}$ at $β_{c}$ will also be larger for your system, both factors in the denominator contribute to making your reliable reweighting range narrower.

Finally, a note on error analysis: The numerator and denominator in the reweighting formula are highly correlated. Standard error propagation formulas will give incorrect (and usually overestimated) errors. The statistically sound way to compute errors for reweighted data is to use resampling methods like Jackknife or Bootstrap, as mentioned in the resources.

5. Generalizations

The reweighting technique is remarkably general. It is not limited to temperature. You can reweight with respect to any parameter that appears linearly in the Hamiltonian. A common and powerful application is reweighting in an external field. For the Ising model, the energy in a field $h$ is $H^{'} = H_{0} - h M$ , where $M$ is the total magnetization.

The reweighting formula for changing both temperature and field from a simulation at $(β_{0}, h_{0})$ becomes:

⟨ O ⟩_{β^{'}, h^{'}} \approx \frac{\sum _{i = 1}^{N} O _{i} e ^{- (β^{'} - β_{0}) E_{i} + (β^{'} h^{'} - β_{0} h_{0}) M_{i}}}{\sum _{i = 1}^{N} e ^{- (β^{'} - β_{0}) E_{i} + (β^{'} h^{'} - β_{0} h_{0}) M_{i}}}

This allows you to map out the equation of state $M (T, h)$ from a single simulation performed at $h = 0$ .

Conclusion

Today, we've added a crucial data analysis tool to our computational toolkit. By understanding and applying histogram reweighting, we can multiply the value of every CPU hour spent on simulations.

Key Takeaways:

Histogram reweighting allows us to calculate thermodynamic observables over a range of temperatures using data from a single simulation.
The core of the method is the formula $⟨ O ⟩_{β^{'}} \approx \frac{\sum O _{i} e ^{- Δ β E_{i}}}{\sum e ^{- Δ β E_{i}}}$ , which is applied to a time series of measured energies and observables.
The technique works by re-weighting the Boltzmann probability of each configuration measured at $β_{0}$ to its new probability at $β^{'}$ .
The valid reweighting range is limited by statistics and shrinks as system size increases, especially near a critical point.
The method can be generalized to reweight in any parameter of the Hamiltonian, such as an external magnetic field.

Preview of the Next Lesson:

We now have the tools to (1) efficiently generate data for a system of size $L$ near $T_{c}$ and (2) use reweighting to precisely locate the peaks of quantities like susceptibility $χ_{L} (T)$ and specific heat $C_{V, L} (T)$ . The next logical question is: how do these peak locations and values change as we vary the system size $L$ ? The answer lies in our next topic, where we will learn to state the finite-size scaling ansatz for thermodynamic observables like magnetization and susceptibility. This will form the bridge between our simulation data and the universal critical exponents that characterize a phase transition.