System–Bath Interaction Models

• System–bath interaction models are frameworks that describe how a finite system exchanges energy with a complex environment, emphasizing relaxation and fluctuation dynamics.
• They employ Hamiltonians divided into system, bath, and interaction components to rigorously analyze energy transfer and statistical behaviors in open systems.
• These models reveal that system size drives nonextensive statistics and induced interactions, impacting energy distributions and collective phenomena in mesoscopic regimes.

A system–bath interaction model describes the dynamical coupling between a finite (or small) system and an environment (“bath”) consisting of many degrees of freedom, typically modeled as harmonic oscillators or similar microscopic subsystems. These frameworks underpin the theoretical treatment of energy exchange, equilibration, and statistical properties in open systems. Originating in classical and quantum statistical mechanics, system–bath models formalize how a system exchanges energy and information with its environment, providing a rigorous foundation for nonequilibrium thermodynamics, relaxation theory, and the emergence of statistical behaviors in small or mesoscopic systems.

1. Generalized System–Bath Hamiltonians

Core system–bath interaction models are formalized by explicitly specifying the total Hamiltonian as a sum of three contributions: $H = H_S + H_B + H_I$ where:

• $H_S$ is the Hamiltonian of the system with $N_S$ degrees of freedom,
• $H_B$ represents the bath, very often modeled as $N_B$ independent or coupled harmonic oscillators,
• $H_I$ is the interaction term, capturing the coupling between system and bath degrees of freedom.

In the classical regime, a widely adopted model is: $$\begin{align*} H_S &= \sum_{k=1}^{N_S} \left[\frac{P_k^2}{2M} + V(Q_k)\right] - f(t)\sum_{k=1}^{N_S}Q_k,\ H_B &= \sum_{n=1}^{N_B} \left[\frac{p_n^2}{2m} + \frac{1}{2}m\omega_n^2 q_n^2\right],\ H_I &= \frac{1}{2}\sum_{k=1}^{N_S}\sum_{n=1}^{N_B} c_{kn}(Q_k - q_n)^2, \end{align*}$$ where $c_{kn}$ are coupling constants, chosen to scale as $c_0/(N_S N_B)$ to maintain finite net interaction for large bath size (Hasegawa, 2010).

Such models encompass various system types ($N_S$ particles, ideal gas, harmonic oscillators), and allow for generalization to more complex baths, including coupled linear oscillator chains with different boundary conditions.

2. Dynamics of Energy Exchange and Recurrence

Direct numerical simulations of the full set of coupled differential equations reveal that the energy in the system, per particle, exhibits two separated timescales:

• Fast, large-amplitude oscillations of the energy $u_S(t)$,
• A slowly modulated envelope dictating the global energy exchange between system and bath.

For finite baths ($N_B \sim 10–1000$) and not too large systems ($N_S \sim 1–10$), no net irreversible dissipation emerges over observation periods—energy returns to the system due to Poincaré recurrence (Hasegawa, 2010). This is in contrast to the Markovian limit (infinite bath), where energy loss is monotonic and irreversible.

The magnitude of energy fluctuations is contingent on the system size: $N_S=1$ shows large relative energy swings, while $N_S>1$ results in more stable ensemble-averaged energy trajectories.

3. Stationary Distributions and Nonextensive Statistics

A central finding concerns the stationary (time-averaged) energy distribution $f_S(u)$ of the small system. The qualitative features are:

• For $N_S=1$, $f_S(u)$ decays approximately exponentially,
• For $N_S > 1$, $f_S(u)$ develops a well-defined peak near the average bath energy.

The form of $f_S(u)$ is well described by a gamma or $\Gamma$ distribution under Boltzmann-Gibbs assumptions: $$f_S(u) \propto u^{a-1} e^{-bu}, \;\;\; a = N_S, \;\;\; b = N_S\beta,$$ where $\beta$ is the inverse temperature. To incorporate non-Gaussian “fat-tailed” features observed, a $q$–$\Gamma$ distribution (employing the $q$-exponential $e_q^{-bu}$) is used,

$f_S(u) = \frac{1}{Z_q} u^{a-1} e_q^{-bu}.$

Here, the entropic index $q$ quantifies the nonextensivity and is assigned based on the system size (superstatistical approach): $q = 1 + \frac{1}{N_S}.$ Alternatively, a microcanonical approach assigns $q = 1 - 1/(N_B - 1)$, but simulations show little dependence on $N_B$, favoring the system-sized-based $q$ (Hasegawa, 2010).

4. Comparative Statistical Approaches: SSA vs. MCA

The superstatistical approach (SSA) postulates fluctuations in inverse temperature (local nonequilibrium), leading directly to a $q$–$\Gamma$ distribution with $q>1$ controlled by $N_S$. The microcanonical approach (MCA) ties $q$ to the bath size and typically results in $q<1$.

Simulations decisively show that $f_S(u)$ is dictated primarily by $N_S$, not $N_B$, thus supporting the SSA for classical open systems of finite size. This underscores that the effective nonextensivity and “fat-tailed” statistics arise from the small system structure rather than the details of the bath (Hasegawa, 2010).

5. Robustness and Bath Engineering

Robustness of these statistical features is demonstrated across:

• Ideal gas systems (with $V(Q_k)=0$), where $f_S(u)$ shows analogous behavior but with rescaled exponents due to reduced degrees of freedom.
• Baths modeled as coupled oscillator chains (periodic or fixed-end boundary conditions), where system distributions $f_S(u)$ remain essentially unchanged even though bath correlations and dynamics are altered (Hasegawa, 2010).

This indicates that, for a broad class of classical systems with quadratic or linear coupling, the precise structure and correlations within the bath have only minor consequence on the system's stationary statistics.

6. Implications for Small and Mesoscopic Systems

The generalized $(N_S+N_B)$ framework uncovers emergent interaction mechanisms among system particles, mediated by the bath: for $N_S>1$, an induced “superexchange” term arises in the effective Langevin equations, absent in single-particle models. This is critical for correct modeling of mesoscopic and nanoscopic systems where these induced interactions can impact collective behavior and transport.

Furthermore, the dominance of system-derived nonextensivity implies that non-Boltzmann statistics (with $q>1$) naturally emerge as a property of finite-sized systems. This becomes central in understanding energy redistribution and the statistical mechanics of few-body open systems, including energy transport and thermalization in molecular, nanoscale, or small condensed-phase environments.

7. Summary Table

Feature	Classical $(N_S, N_B)$ Model (Hasegawa, 2010)	Statistical Approach
Stationary $f_S(u)$	Exponential for $N_S=1$; peaked for $N_S>1$	$\Gamma$ or $q$–$\Gamma$
Fluctuation Scaling	Rapid, with slow envelope; amplitude $\sim N_S^{-1}$	SSA: $q=1+1/N_S$
Bath Size Dependence	Weak; $N_S$ dominates	MCA: $q=1-1/(N_B-1)$
Bath Correlations	Minimal effect on $f_S(u)$	Robust to bath structure
Induced Interactions	Present for $N_S>1$	Absent for $N_S=1$

These findings establish a precise framework for modeling and understanding energy fluctuations, emergent statistical distributions, and system-size-driven nonextensivity in small classical systems coupled to finite environments. They provide foundational guidance for both theoretical and applied modeling of open mesoscopic systems (Hasegawa, 2010).