Stochastic Thermodynamics with Internal Variables

• Stochastic Thermodynamics with Internal Variables is a framework that explicitly incorporates hidden microstates into the thermodynamic description to capture nonequilibrium dynamics.
• It rigorously defines energy, entropy, and free energy balances at both single-trajectory and ensemble levels, accommodating strong system-environment coupling without weak-coupling assumptions.
• The formulation enforces local detailed balance and fluctuation theorems, underpinning practical applications such as enzyme kinetics, molecular motors, and soft-matter systems.

Stochastic thermodynamics with internal variables generalizes classical and stochastic thermodynamics by explicitly incorporating hidden mesoscopic or microscopic degrees of freedom—referred to as internal variables—into the thermodynamic state description. The framework provides a rigorous mathematical apparatus for systems where observable behavior emerges from averaging over fast-relaxing microstates, and is particularly suited for enzymes, molecular motors, soft-matter systems, and small devices where strong coupling to the environment, memory effects, or nonequilibrium driving are essential. It delivers energy, entropy, and free energy balances at the single-trajectory and ensemble levels, ensures consistency with the first and second laws, and uniquely constrains the stochastic kinetics to maintain thermodynamic structure even far from equilibrium.

1. Definition and Role of Internal Variables

Internal variables are mesoscopic or microscopic degrees of freedom that are not directly controlled via thermal or mechanical constraints but are essential to the system's nonequilibrium behavior. In the context of single enzymes or motors, they include conformational states and binding configurations; in phase-space kinetic equations, they can be collision angles or velocities; in macroscopic thermodynamic models, they may be coarse-grained order parameters or contact temperatures (1012.3119, Muschik, 2017). Formally, the total microstate $\xi$ is partitioned into classes or basins labeled by discrete states $n$, which are characterized by distributions over many rapidly equilibrating microstates (e.g., conformational substates, internal molecular arrangements, occupancy by bound solutes). These internal variables are “integrated out” to define effective thermodynamic state functions—internal energy $E_n$, intrinsic (configurational) entropy $S_n^{\text{enz}}$, and free energy $F_n$—which capture the physics of the underlying microstates (1012.3119).

Crucially, the thermodynamics with internal variables does not require weak system-bath coupling: the system can be strongly coupled to its environment, and the time-scale separation between internal degrees of freedom (fast) and nonequilibrium transitions (slow) underpins the Markovian mesoscopic description adopted by the framework (1012.3119, Ding et al., 2021, Xing, 30 Jun 2025).

2. Trajectory-Level Thermodynamic Balances

At the trajectory level, thermodynamic quantities are defined for individual stochastic paths comprising jumps or transitions between mesoscopic/metastable states (e.g., between enzyme conformations, or between coarse-grained attractors). Each transition, $m \to n$, is associated with:

• Heat Exchange: $q = -\Delta E = -(E_n - E_m)$, equating the heat released/absorbed to the negative change in the internal energy (including contributions from internal variables, regardless of coupling strength).
• Work and Chemical Work: For molecular motors, mechanical work is $f d_\rho$ per transition of length $d_\rho$ under force $f$. Chemical work is included through potential differences $\Delta \mu_\rho$, stemming from binding or release of solute molecules.
• Stochastic Entropy Change: For an ensemble trajectory, the stochastic entropy at time $t$ is $s(t) = -\ln p_n(t)$, where $p_n(t)$ is the time-dependent probability to be in state $n$ (1012.3119).

The first law is enforced as an energy-conserving balance: $\Delta E = -q$ Incorporating mechanical work and chemical work, when appropriate, leads to extended expressions, e.g.,

$$q_\rho = f d_\rho - \Delta E_\rho + T (\partial_T \Delta S^{\text{enz}}_\rho)$$

where $T$ denotes temperature and $S^{\text{enz}}_\rho$ tracks changes in the intrinsic entropy.

3. Entropy Production and the Second Law

Total entropy production for a single transition in the ensemble is decomposed as: $$\Delta S^{\text{tot}} = \Delta S^{\text{med}} + \Delta S^{\text{enz}} + \Delta s$$ where:

• $\Delta S^{\text{med}} = q/T$ is the entropy change in the medium,
• $$\Delta S^{\text{enz}} = S_n^{\text{enz}} - S_m^{\text{enz}}$$ is the difference in intrinsic entropy,
• $\Delta s = -\ln[p_n(t)/p_m(t)]$ is the change in stochastic entropy (1012.3119).

Non-negativity of $\langle \Delta S^{\text{tot}} \rangle$ enforces the second law on the ensemble level. This sum must be used: omitting the stochastic entropy would erroneously permit violations of the second law during measurement. Furthermore, fluctuation theorems hold: $$\langle \exp(-\Delta S^{\text{tot}}/k_B) \rangle = 1$$ implying exponential suppression of negative total entropy fluctuations (1012.3119).

4. Local Detailed Balance and Transition Kinetics

A fundamental outcome is the local detailed balance (LDB) condition, which uniquely constrains the stochastic kinetics: $$\frac{w_\rho^+}{w_\rho^-} = \begin{cases} \exp[ -\beta(\Delta F^{\text{enz}}_\rho + \Delta \mu_\rho) ] & \text{(enzymatic)} \ \exp[ -\beta(\Delta F^{\text{enz}}_\rho + \Delta \mu_\rho - f d_\rho) ] & \text{(motor, with force)} \end{cases}$$ This ensures that the transition rates are consistent with the underlying thermodynamic potentials defined via internal variables. The exponential form of the LDB guarantees that the global detailed balance is satisfied in equilibrium and that entropy production remains non-negative out of equilibrium (1012.3119).

In coarse-grained or multiscale systems, thermodynamic consistency across levels is preserved: integrating out fast (internal) variables preserves the Helmholtz free energy, entropy, and entropy production, provided time-scale separation holds (1103.3306). Thus, describing the system as a Markov jump process with empirically determined rates does not lose thermodynamic information.

5. Ensemble and System-Level Thermodynamics

Upon ensemble averaging, internal variables continue to play a central role in state functions and the evolution of probability distributions. The entropy,

$S(t) = -k_B \sum_n p_n(t) \ln p_n(t)$

captures stochasticity in state occupation, while the internal energy and free energy are averages over their state-dependent values, which include contributions from internal variables (e.g., substrate binding, conformational entropy).

Master equations for discrete states or Fokker–Planck equations for continuous variables are formulated with internal-variable-resolved transition rates. The time evolution explicitly ensures the monotonic decrease of free energy and non-negativity of entropy production (Tomé et al., 2015).

6. Mathematical Representation and Key Formulae

Important equations include:

• State free energy for bound molecules: $$F_n^{\text{enz}}(\{c_i\}) = F(\{N_i\}) + \sum_i r_i \mu_i$$
• First law for a transition: $\Delta E = E_n - E_m = -q$
• Stochastic entropy: $s(t) = -\ln p_n(t)$
• Entropy production increment upon a transition: $\Delta s_\rho(t) = -k_B \ln [p_{n'}(t)/p_n(t)]$
• Local detailed balance: $$\frac{w_\rho^+}{w_\rho^-} = \exp[-\beta (\Delta F_\rho^{\text{enz}} + \Delta \mu_\rho - f d_\rho)]$$
• Fluctuation theorem (integral form): $\langle e^{-\Delta S^{\text{tot}}/k_B} \rangle = 1$

7. Assumptions, Limitations, and Generalizations

The framework relies on clear time-scale separation: microstates within a mesostate equilibrate rapidly relative to the slow transitions between mesostates. No assumption of weak coupling to the bath is needed; strong system-bath interactions may be present (1012.3119, Ding et al., 2021, Xing, 30 Jun 2025). It is not dependent on mass-action kinetics or ideal solution behavior, and makes no restrictive assumptions on the explicit form of transition rates, other than LDB.

A potential limitation is that, given only a master equation and observed rates, the reverse determination of state thermodynamic properties (the "inverse problem") is ill-posed without detailed microscopic knowledge. Additionally, in dynamic nonequilibrium conditions (e.g., time-dependent force or solute concentrations), state functions become explicitly time-dependent.

Local detailed balance and entropy production formulation extend to continuous systems, coarse-grained Markov processes, and multiple scales provided time-scale separation persists (1103.3306, Smith, 2020). The approach has also been generalized to systems with memory (non-Markovianity) and internal variable-driven memory kernels (Cockrell et al., 2022), and to strong coupling and microcanonical ensemble frameworks, where the bath's Boltzmann entropy and the heat as the negative change of bath energy are precisely defined (Xing, 30 Jun 2025).

8. Practical and Theoretical Impact

This framework underlies the modeling of single-molecule biophysics, chemical reaction networks, phase-transforming soft materials, micromagnetic systems, and active matter, providing consistent definitions of work, heat, and entropy production in the presence of internal variables. It unifies trajectory-level thermodynamics (fluctuation theorems), ensemble-level balances (first/second law), and the construction of reduced models or data-driven coarse-graining. It highlights the necessity of including stochastic entropy and LDB for physical consistency in synthetic and biological nanodevices and clarifies the critical role of hidden (internal) degrees of freedom in nonequilibrium dynamics.

A summary of the core contributions and formulae is provided below:

Concept	Mathematical Expression	Notes
State free energy	$$F_n^{\text{enz}}(\{c_i\}) = F(\{N_i\}) + \sum_i r_i \mu_i$$	Includes chemical potentials of bound solutes
First law in transitions	$\Delta E = E_n - E_m = -q$	Energy balance including internal variables
Stochastic entropy	$s(t) = -\ln p_n(t)$	Along individual trajectory
Entropy production	$$\Delta S^{\text{tot}} = \Delta S^{\text{med}} + \Delta S^{\text{enz}} + \Delta s$$	Ensures second law on average
Local detailed balance (LDB)	$$\frac{w_\rho^+}{w_\rho^-} = \exp[-\beta (\Delta F^{\text{enz}}_\rho + \Delta \mu_\rho - f d_\rho)]$$	Governs kinetic rates
Fluctuation theorem	$$\langle \exp(-\Delta S^{\text{tot}}/k_B) \rangle = 1$$	Implies non-negative entropy production on average

This framework establishes that internal variables—representing hidden or coarse-grained microscopic structure—are fundamental to a thermodynamically consistent description of stochastic mesoscopic and nanoscopic systems, ensuring exact first- and second-law balances and the emergence of fluctuation relations across a wide range of far-from-equilibrium phenomena.