Analytical measure of temperature for nonlinear dynamical systems

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Nonlinear Dynamics

Background:

• Measuring temperature in microcanonical ensembles of nonlinear dynamical systems is computationally intensive.
• Existing methods often rely on time averages, which can be slow and resource-demanding.

Purpose of the Study:

• To develop an analytical approach for determining the temperature of nonlinear dynamical systems in the microcanonical ensemble.
• To validate the hypothesis of ensemble equivalence in these systems.
• To provide a computationally efficient alternative to existing methods.

Main Methods:

• Utilizing self-consistent phonon theory to analytically derive temperature from internal energy density.
• Applying the approach to FPU-β and ϕ⁴ lattice models.
• Comparing analytical results with time averages from phase space trajectories and thermostat temperatures.

Main Results:

• The analytical approach successfully measures temperature in the microcanonical ensemble for the studied models.
• Results are consistent with traditional methods (time averages) and thermostat temperatures.
• Finite size effects were quantified, and computational performance improves with system size.

Conclusions:

• The proposed analytical method provides an efficient and accurate way to measure temperature in nonlinear dynamical systems.
• The study validates the hypothesis of ensemble equivalence for these systems.
• This approach offers significant advantages over time-consuming computational methods.