Time Evolution of Heat Conduction in a Generalized Model of Brownian Motion

Abstract: We investigate the properties of heat conduction in a network of harmonic oscillators interacting with heat baths, described by a generalized model of Brownian motion. This model includes noise and dissipation terms in both the momentum and position equations. This generalization is motivated by the requirement of consistency with the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. Because standard definitions of heat current based on velocity become mathematically inconsistent in this framework, we derive an analytical expression for the steady-state heat flow based on an extended framework of stochastic energetics. We confirm that Fourier's law (linear thermal response) is satisfied and that the model naturally captures microscopic thermal boundary resistance, analogous to Kapitza resistance. This demonstrates that our generalized model functions as a valid phenomenological framework for simulating non-equilibrium processes, marking a crucial step toward a unified formulation of stochastic and quantum thermodynamics. Furthermore, we analyze the time evolution of heat conduction by numerically solving the corresponding differential equations for the correlation functions. Unlike standard Brownian motion, the generalized model generates continuous and nowhere differentiable trajectories for both momentum and position (as is characteristic of overdamped dynamics). Finally, we show that the heat current exhibits characteristic transient behavior when the inter-particle interaction is switched on. Specifically, an instantaneous heat flow emerges, whose direction is strictly governed by whether the interaction is attractive or repulsive, significantly differing from the predictions of the standard model.

• The paper introduces a generalized model by incorporating stochastic fluctuations in both position and momentum, yielding a comprehensive formulation of heat conduction.
• The methodology employs coupled harmonic oscillators and stochastic differential equations to derive Fourier’s law and analyze transient phenomena.
• Numerical and analytical results confirm the emergence of macroscopic heat conduction properties, including Kapitza resistance at thermal interfaces.

Time Evolution of Heat Conduction in a Generalized Model of Brownian Motion

Introduction and Motivation

The study "Time Evolution of Heat Conduction in a Generalized Model of Brownian Motion" (2606.08839) addresses the derivation of macroscopic heat conduction laws from microscopic dynamics, focusing on achieving a thermodynamically consistent framework that naturally reconciles with quantum open system descriptions. The work extends the classical Brownian motion paradigm by incorporating fluctuation and dissipation directly into both position and momentum equations, ensuring compatibility with the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation and the CPTP condition required for quantum master equations.

This generalized approach entails that position and momentum exhibit continuous but nowhere differentiable stochastic trajectories, a qualitative departure from standard models where only momentum is subject to stochastic influences. The paper rigorously studies whether such modified microscopic dynamics sustain macroscopic principles such as Fourier’s law and whether the approach faithfully encapsulates non-trivial phenomena like Kapitza resistance at thermal interfaces.

Theoretical Framework

Building on a network of linearly coupled harmonic oscillators interacting with individual heat reservoirs, the authors use a set of stochastic differential equations where both $\tilde{q}_i$ and $\tilde{p}_i$ evolve under independent Wiener processes. The dissipation coefficients $\gamma_{q_i}$ (position) and $\gamma_{p_i}$ (momentum) encode the balance between fluctuation and relaxation mechanisms. This setting interpolates between the traditional Brownian case ($\gamma_{q_i} \to 0$) and the fully generalized model.

Heat is consistently defined via stochastic energetics, including both the position and momentum noise contributions. This extension is essential due to the breakdown of the conventional velocity-momentum correspondence when noise acts on position. The heat current expressions thus ensure compliance with the first and second laws at the trajectory level, even for nowhere-differentiable stochastic paths.

Emergence of Fourier’s Law and Steady-State Analysis

The analysis focuses first on the $N=2$ oscillator system, providing tractable algebra and facilitating detailed analytical treatment. Linearization and dimensional reduction reveal that, in steady state, the heat current $J_1^\text{ss}$ between the baths is strictly proportional to the temperature difference $\Delta T$:

$J_1^\text{ss} = \kappa \Delta T$

where an explicit analytical expression is derived for the thermal conductivity $\kappa$ as a function of model parameters. The solution incorporates finite $\tilde{p}_i$0, confirming that Fourier-type linear thermal response persists within the generalized framework.

For the non-interacting limit ($\tilde{p}_i$1), equipartition is satisfied and each oscillator thermalizes with its respective bath. For nonzero interaction, the effective temperature discontinuity between oscillators and baths naturally arises, embodying microscopic thermal boundary resistance—a robust manifestation of Kapitza resistance observed experimentally.

Stochastic Trajectories: Differentiability and Dynamics

The inclusion of position noise induces dramatic structural changes in the system's stochastic trajectories. For $\tilde{p}_i$2, only momentum exhibits jagged, nowhere-differentiable paths, while position remains smooth. When $\tilde{p}_i$3, both momentum and position are rendered continuous but nowhere differentiable.

Figure 1: Trajectories obtained by solving the generalized stochastic differential equations—momentum (top) and position (bottom) for $\tilde{p}_i$4 (left, smooth position) and $\tilde{p}_i$5 (right, both zigzag).

This feature is crucial, as the stochastic calculus no longer supports the simplistic association of velocity with momentum, making RLL-type current definitions (force-velocity products) ill-posed.

Transient Regimes and Non-equilibrium Effects

Numerical integration of the correlation function evolution equations exposes rich transient dynamics upon sudden activation of interparticle interaction:

• For standard Brownian motion ($\tilde{p}_i$6), immediately after interaction onset, heat backflows from both baths due to a sudden increase in system potential energy.
• With $\tilde{p}_i$7, the heat current exhibits a finite instantaneous response, whose sign is dictated by the nature of the interaction—negative for attractive, positive for repulsive. This “instantaneous jump” arises from direct stochastic energy transfer mediated by position noise, distinct from standard models.

Figure 2: The time evolution of the heat current $\tilde{p}_i$8, system energy derivative $\tilde{p}_i$9, and heat current $\gamma_{q_i}$0 for standard Brownian motion ($\gamma_{q_i}$1) with various temperature biases.

Figure 3: The time evolution of the heat current and energy derivative for the generalized model with $\gamma_{q_i}$2 demonstrates the initial heat current jump and the subsequent relaxation to steady-state.

As $\gamma_{q_i}$3 increases, the transient minimum in $\gamma_{q_i}$4 disappears, yielding monotonic relaxation. The theoretical dependence of initial currents on interaction sign and strength is quantitatively accounted for by the analytically derived expressions.

Figure 4: The time evolutions of $\gamma_{q_i}$5 for increasing $\gamma_{q_i}$6, showing monotonic approach to steady state without the minima observed for smaller $\gamma_{q_i}$7.

For repulsive couplings, the energy change and heat flows reverse direction as predicted; the framework captures these energy-budget determinants in full generality.

Figure 5: Heat current, energy rate, and counter-current plots for repulsive coupling ($\gamma_{q_i}$8), demonstrating energy absorption at $\gamma_{q_i}$9.

Comparison with Standard Approaches and Implications

By referencing the RLL model, which assumes smooth velocity fields and defines heat current as force-velocity products, the article stresses the breakdown of these constructs in the generalized setting. The generalized model’s definition—via stochastic energetics—remains robust under nowhere-differentiable paths and upholds rigorous first and second law consistency even when force-velocity products become mathematically ill-defined.

Implications and Future Directions

The work demonstrates that generalized Brownian motion, when endowed with stochasticity and dissipation in both position and momentum, is fully compatible with linear-response phenomena and interfacial resistances observed in real systems. Importantly, the formulation sustains strong consistency with quantum thermodynamic requirements. The approach also resolves known deficiencies of overdamped stochastic models (e.g., divergence of conductivity in the strong-coupling limit).

The framework is positioned as a benchmark for exploring quantum-to-classical correspondence in thermodynamic transport, with implications for ongoing debates between global and local master equation implementations in open quantum systems. The natural emergence of a global dynamical structure in the quantized version underscores the validity of this approach for modeling realistic system-bath interplay.

A pivotal open question concerns the persistence of anomalous transport (e.g., ballistic vs. diffusive behavior) in extended one-dimensional chains ($\gamma_{p_i}$0) with non-zero positional dissipation. The presence of position noise could act as an intrinsic scattering mechanism, potentially suppressing long-range correlations and reestablishing normal, size-independent conductivity.

Conclusion

This paper provides a mathematically rigorous and physically consistent framework for analyzing heat conduction in systems described by generalized Brownian motion, ensuring compatibility with both stochastic thermodynamics and open quantum system theory. By analytically and numerically investigating both steady-state and transient phenomena, the work validates the emergence of Fourier’s law and microscopic interfacial resistance beyond the reach of standard Brownian models. The approach establishes essential foundations for studying mesoscopic and quantum-classical nonequilibrium phenomena and suggests promising future directions for investigating anomalous and normal transport in extended systems.