Jarzynski Two-State Quench

Worked Example

Consider a system with Inverse temperature beta = 1, Initial excited energy ei = 1, and Final excited energy ef = 4. The ground state energy is always 0.

// Calculate initial and final partition functions Z_init = 1 + exp(-beta * ei) = 1 + exp(-1 * 1) = 1 + 0.367879... = 1.367879... Z_final = 1 + exp(-beta * ef) = 1 + exp(-1 * 4) = 1 + 0.018315... = 1.018315... // Free-energy difference DeltaF = -(1/beta) * ln(Z_final / Z_init) = -1 * ln(1.018315 / 1.367879) = -ln(0.744448...) = 0.295111... // Probabilities of being in each state initially p_ground  = 1 / Z_init = 1 / 1.367879 = 0.731058... p_excited = exp(-beta * ei) / Z_init = 0.367879 / 1.367879 = 0.268941... // Work done during the instantaneous quench W_ground  = 0 - 0 = 0 W_excited = ef - ei = 4 - 1 = 3 // Exponential work average E[exp(-beta * W)] = p_ground * exp(-beta * W_ground) + p_excited * exp(-beta * W_excited) = 0.731058 * exp(-1 * 0) + 0.268941 * exp(-1 * 3) = 0.731058 * 1 + 0.268941 * 0.049787 = 0.731058 + 0.013390 = 0.744448... // Jarzynski Equality E[exp(-beta * W)] = Z_final / Z_init 0.744448... = 1.018315... / 1.367879... = 0.744448...

The calculation confirms the Jarzynski Equality holds, and the result is independent of the path taken, relying only on the initial and final equilibrium states.

Experiment

You can interactively explore the relationship between the exponential work average and the free-energy difference in the Jarzynski Two-State Quench module.

1. Change the Inverse temperature to observe how it affects the initial state probabilities and the resulting exponential weighting.
2. Adjust the Initial excited energy and Final excited energy to simulate different instantaneous quench scenarios.
3. Observe that while the Mean work is generally greater than or equal to the Free-energy difference (in accordance with the Second Law of Thermodynamics and Jensen's inequality), the Exponential work average perfectly matches the Partition-function ratio.
4. Notice that negative work and free-energy changes are possible for downward quenches (where the final energy is lower than the initial energy), but the mean dissipated work remains non-negative.

Limitations

This module calculates exact ensemble values for a simplified two-state system. It assumes an instantaneous quench, meaning the energy levels change infinitely fast, so the system does not have time to transition between states during the process itself (no work is done via state transitions, only via changes in the energy levels). It does not represent a finite-sample free-energy estimate, nor does it claim that every individual realization of work exceeds the free-energy difference.

References

The equality was first introduced by Christopher Jarzynski in 1997. The following resources detail its derivation and implications for non-equilibrium systems:

• Nonequilibrium Equality for Free Energy Differences - The original Physical Review Letters paper by C. Jarzynski introducing the equality.
• Nonequilibrium equality for free energy differences (arXiv) - The preprint version of the paper detailing the derivation and implications of the Jarzynski equality.