Entropy production as correlation between system and reservoir (0908.1125v1)

Abstract: We derive an exact (classical and quantum) expression for the entropy production of a finite system placed in contact with one or several finite reservoirs each of which is initially described by a canonical equilibrium distribution. Whereas the total entropy of system plus reservoirs is conserved, we show that the system entropy production is always positive and is a direct measure of the system-reservoir correlations and/or entanglements. Using an exactly solvable quantum model, we illustrate our novel interpretation of the Second Law in a microscopically reversible finite-size setting, with strong coupling between system and reservoirs. With this model, we also explicitly show the approach of our exact formulation to the standard description of irreversibility in the limit of a large reservoir.

• The paper proposes an exact formulation of entropy production using quantum relative entropy to capture system-reservoir correlations that guarantees non-negative values.
• It demonstrates that even under strong coupling and finite reservoirs, the derived entropy production remains positive, validating the approach for non-equilibrium dynamics.
• Numerical simulations on a two-level spin system illustrate the method’s robustness compared to standard definitions, providing key insights for mesoscopic thermodynamics.

This paper (0908.1125) addresses a fundamental problem in statistical mechanics: deriving the Second Law of Thermodynamics from the underlying time-reversible microscopic dynamics. While the total entropy of a closed system (system + reservoirs) remains constant according to Liouville's theorem, the entropy of an open system (just the system) can change. The challenge is to define entropy production for this open system in a way that is always non-negative ($\Delta_i S \geq 0$), even for finite systems strongly coupled to finite reservoirs, without resorting to the standard assumptions of infinite reservoirs or weak coupling.

The authors propose an exact definition for the entropy production of a finite system interacting with one or more finite reservoirs, assuming the reservoirs are initially in canonical equilibrium and are uncorrelated with the system. The initial state of the total system $\rho(0)$ is a product state of the system's initial state $\rho_s(0)$ and the reservoirs' thermal equilibrium states $\rho_r^{\rm eq}$: $\rho(0) = \rho_s(0) \prod_r \rho_r^{\rm eq}$. The system's entropy is defined as the von Neumann entropy $S(t) = -\mathrm{Tr}_s \rho_s(t) \ln \rho_s(t)$, where $\rho_s(t)$ is the reduced density matrix obtained by tracing out the reservoir degrees of freedom from the total density matrix $\rho(t)$.

By exploiting the time-invariance of the total system's von Neumann entropy, the authors rigorously show that the change in the system's entropy, $\Delta S(t) = S(t) - S(0)$, can be exactly decomposed into two terms:

$\Delta S(t) = \Delta_i S(t) + \Delta_e S(t)$

where $\Delta_e S(t)$ is the entropy flow from the reservoirs to the system, given by:

$$\Delta_e S(t) = - \sum_r \beta_r (\langle H_r \rangle_t - \langle H_r \rangle_0)$$

This term is directly related to the heat exchange with the reservoirs. The crucial term is the entropy production, $\Delta_i S(t)$, which is defined as the quantum relative entropy between the total system's state $\rho(t)$ and a hypothetical decorrelated state $\rho_s(t) \prod_r \rho_r^{\rm eq}$:

$$\Delta_i S(t) \equiv D[\rho(t)||\rho_s(t)\prod_r \rho_r^{\rm eq}]$$

The quantum relative entropy $$D[\rho||\rho'] = \mathrm{Tr}(\rho \ln \rho) - \mathrm{Tr}(\rho \ln \rho')$$ is a well-known measure of the "distance" between two density matrices. A key property of relative entropy is that it is always non-negative, $D[\rho||\rho'] \geq 0$, and is zero only if $\rho = \rho'$.

This definition leads to the central result: the entropy production $\Delta_i S(t)$ is always positive, $\Delta_i S(t) \geq 0$. This positivity is interpreted as a direct measure of the correlations and/or entanglement that develop between the system and the reservoirs over time. The state $\rho(t)$ evolves unitarily from the initially decorrelated state, developing correlations. The quantity $\rho_s(t) \prod_r \rho_r^{\rm eq}$ represents the state where these correlations have been "discarded" by considering only the marginal system state and assuming the reservoirs remain in their initial thermal state. The relative entropy quantifies how "far" the true correlated state is from this decorrelated product state. The positivity of $\Delta_i S(t)$ implies that the formation of system-reservoir correlations contributes a negative entropy term to the total system, precisely balancing the apparent entropy increase in the system and reservoirs when correlations are neglected.

The paper connects this entropy production to thermodynamic quantities for a driven system in contact with a single reservoir. Defining work $W(t)$ and the system's energy change $\Delta U(t)$, they show the First Law $\Delta U(t) = W(t) + Q(t)$ holds (where $Q(t)$ is heat absorbed). Introducing a nonequilibrium free energy $\Delta F(t) = \Delta U(t) - T \Delta S(t)$, they recover the standard thermodynamic form for irreversibility:

$T \Delta_i S(t) = W(t) - \Delta F(t) \geq 0$

This relation is shown to be exact, even for strong coupling and finite systems.

The authors contrast their definition $\Delta_i S(t)$ with an alternative definition $\Delta_i \bar{S}(t)$ commonly used in open quantum system theory [Breuer02], which is defined based solely on the system's density matrix and its equilibrium state. While $\Delta_i \bar{S}(t)$ is positive in the weak coupling/large reservoir limit where the system dynamics is Markovian and the system relaxes to a stationary equilibrium state, they show that for finite systems and strong coupling, $\Delta_i \bar{S}(t)$ can become negative, which is physically problematic for entropy production. The difference between the two definitions lies precisely in the interaction energy term.

To illustrate their findings, the authors analyze an exactly solvable quantum model: a two-level spin system coupled to an N-level reservoir via a random matrix Hamiltonian. The total Hamiltonian is $$H = \frac{\Delta}{2} \sigma_z + H_r + \lambda \sigma_x R$$, where $\sigma_{x,z}$ are Pauli matrices, $H_r$ is the reservoir Hamiltonian, and $R$ is a Gaussian orthogonal random matrix representing the coupling with strength $\lambda$. They numerically simulate the exact dynamics of the total system for finite $N$ and compare $\Delta_i S(t)$ and $\Delta_i \bar{S}(t)$.

The numerical results show that $\Delta_i S(t)$ calculated using the exact dynamics remains positive at all times, although it displays oscillations and near-recurrences typical of finite-size systems. In contrast, $\Delta_i \bar{S}(t)$ from the exact dynamics becomes negative for small $N$, demonstrating its inadequacy beyond the standard limits. As the reservoir size $N$ increases, both $\Delta_i S(t)$ and $\Delta_i \bar{S}(t)$ converge to the same positive, monotonically increasing function predicted by the Redfield quantum master equation (which is valid in the weak-coupling, large-reservoir limit). This convergence shows how the exact, microscopically reversible dynamics for finite systems approaches the irreversible behavior described by standard open quantum system theory in the thermodynamic limit.

Practical Implications and Implementation:

• Understanding Irreversibility at the Nanoscale: The paper provides a theoretical framework crucial for understanding thermodynamic processes in mesoscopic and nanoscale systems (like quantum dots, molecular junctions, qubits coupled to environments) where the system-bath coupling might be strong and the bath might not be effectively infinite.
• Defining Entropy Production: The proposed definition $$\Delta_i S(t) = D[\rho(t)||\rho_s(t)\prod_r \rho_r^{\rm eq}]$$ offers a rigorous way to quantify irreversibility in non-equilibrium dynamics for finite quantum systems. This is implementable if one can track or compute the total density matrix $\rho(t)$ and the reduced system density matrix $\rho_s(t)$.
• Computational Requirements: Calculating $\rho(t)$ for a composite system (system + reservoir) is generally computationally intensive. If the system Hilbert space has dimension $d_s$ and the reservoir has dimension $d_r$, the total state space dimension is $d_s d_r$. Storing and evolving $\rho(t)$ (a $(d_s d_r) \times (d_s d_r)$ matrix) scales as $(d_s d_r)^2$. For the illustrative model, $d_s=2$ and $d_r=N$. Exact diagonalization or numerical integration of the Liouville equation $i \hbar \dot{\rho} = [H, \rho]$ for the full Hamiltonian is feasible only for relatively small reservoir sizes $N$. Calculating relative entropy also requires matrix logarithm and trace operations.
• Role of Correlations: The definition explicitly highlights that irreversibility, from a microscopic perspective, is tied to the development of correlations/entanglement between the system and its environment. Measuring or quantifying these correlations in experiments could provide insight into the microscopic origins of thermodynamic irreversibility.
• Beyond Weak Coupling: Standard quantum master equations (like Redfield or Lindblad forms) are often derived under approximations like weak coupling and Markovian evolution. The paper's framework is exact and applicable outside these regimes, which is important for systems exhibiting non-Markovian behavior or strong coupling.
• Model Simulation: Implementing the simulation of the spin-GORM model requires:
  • Defining the Hamiltonians $H_s$, $H_r$ (diagonal with specific level spacing), and the interaction $V = \lambda \sigma_x R$. Generating the random matrix $R$.
  • Setting up the initial state $\rho(0) = \rho_s(0) \otimes \rho_r^{eq}$. $\rho_r^{eq}$ is a thermal state $\exp(-\beta H_r)/Z_r$.
  • Solving the Liouville equation $i \dot{\rho}(t) = [H, \rho(t)]$ for $\rho(t)$, e.g., using matrix exponentiation: $\rho(t) = e^{-iHt} \rho(0) e^{iHt}$.
  • Calculating $\rho_s(t) = \mathrm{Tr}_r \rho(t)$.
  • Calculating $$\Delta_i S(t) = \mathrm{Tr}(\rho(t) \ln \rho(t)) - \mathrm{Tr}(\rho(t) \ln(\rho_s(t) \otimes \rho_r^{eq}))$$.
  • Calculating $$\Delta_i \bar{S}(t) = D[\rho_s(0)||\rho_s^{eq}] - D[\rho_s(t)||\rho_s^{eq}]$$, where $\rho_s^{eq} \propto \exp(-\beta H_s)$.

While the paper is primarily theoretical, its rigorous definition of entropy production provides a robust foundation for developing simulation methods and experimental protocols to probe thermodynamic irreversibility in mesoscopic systems, where the effects of finite size, strong coupling, and non-Markovian dynamics are significant and cannot be ignored. The core challenge for practical implementation is obtaining the total system density matrix, which remains computationally demanding for large environments.