A Thermodynamically Consistent Thermal Equilibrium Gaussian White Noise Model for Nonlinear Resistors

Traditional extensions of the Nyquist-Johnson formula for thermal fluctuations in nonlinear dissipative elements have often led to thermodynamically inconsistent models and sparked long-standing debates about the proper interpretation of stochastic differential equations. In this work, we show that it is possible to derive, for each of the main stochastic interpretations, a Gaussian white-noise model for nonlinear dissipative elements at thermal equilibrium that fully complies with the fundamental principles of thermodynamics. The resulting models reproduce the Gibbs (Maxwell-Boltzmann) distribution and ensure zero-mean voltages and currents, thereby resolving the Brillouin paradox and maintaining consistency with the second law of thermodynamics. Furthermore, we demonstrate that these models satisfy additional thermodynamic requirements, including positive entropy production during transients and zero net heat exchange between dissipative elements at equilibrium.