The conjugate-pairing rule for non-Hamiltonian systems

Area of Science:

• Thermodynamics
• Statistical Mechanics
• Computational Physics

Background:

• The Conjugate Pairing Rule (CPR) describes symmetric Lyapunov exponent spectra in certain systems.
• CPR simplifies calculating transport coefficients in dissipative systems by relating Lyapunov exponents to entropy production.
• Sufficient conditions for CPR involve symplectic adiabatic dynamics, but necessary conditions remain unknown.

Purpose of the Study:

• To investigate the necessary conditions for the Conjugate Pairing Rule (CPR) to hold.
• To determine if standard molecular dynamics algorithms for shear viscosity calculation adhere to CPR.
• To explore the relationship between Lyapunov exponents and entropy production in systems violating CPR.

Main Methods:

• Computer simulations were employed to analyze systems satisfying the Conjugate Pairing Rule (CPR).
• The study focused on the standard molecular dynamics algorithm for calculating shear viscosity.
• Analysis involved examining the spectrum of Lyapunov exponents and entropy production.

Main Results:

• Convincing evidence suggests the standard molecular dynamics algorithm for shear viscosity violates CPR, even in the thermodynamic limit.
• Despite violating CPR, the sum of the maximal Lyapunov exponents appears to equal the entropy production per degree of freedom.
• This indicates potential for accurate shear viscosity calculation using the standard algorithm.

Conclusions:

• The necessary conditions for the Conjugate Pairing Rule (CPR) are not fully understood.
• Standard molecular dynamics algorithms for shear viscosity calculation do not satisfy CPR.
• Shear viscosity can likely still be accurately computed by summing the maximal Lyapunov exponents, despite CPR violation.