Approach to equilbrium in nano-scale systems at finite temperatur

We study the time evolution of the reduced density matrix of a system of spin-1/2 particles interacting with an environment of spin-1/2 particles. The initial state of the composite system is taken to be a product state of a pure state of the system and a pure state of the environment. The latter pure state is prepared such that it represents the environment at a given finite temperature in the canonical ensemble. The state of the composite system evolves according to the time-dependent Schr{"{o}}dinger equation, the interaction creating entanglement between the system and the environment. It is shown that independent of the strength of the interaction and the initial temperature of the environment, all the eigenvalues of the reduced density matrix converge to their stationary values, implying that also the entropy of the system relaxes to a stationary value. We demonstrate that the difference between the canonical density matrix and the reduced density matrix in the stationary state increases as the initial temperature of the environment decreases. As our numerical simulations are necessarily restricted to a modest number of spin-1/2 particles ($<36$), but do not rely on time-averaging of observables nor on the assumption that the coupling between system and environment is weak, they suggest that the stationary state of the system directly follows from the time evolution of a pure state of the composite system, even if the size of the latter cannot be regarded as being close to the thermodynamic limit.

💡 Research Summary

The paper investigates how a small quantum system composed of spin‑½ particles reaches thermal equilibrium when coupled to an environment that is also made of spin‑½ particles. The authors prepare the composite system in a product state: the system is in an arbitrary pure state, while the environment is placed in a pure state that mimics a canonical ensemble at a prescribed temperature (T). This “canonical pure state” is generated by applying the operator (\exp(-\beta H_E/2)) to a random vector and normalising, where (\beta=1/k_B T) and (H_E) is the environment Hamiltonian. The total Hamiltonian consists of three parts—(H_S) for the system, (H_E) for the environment, and an interaction term (H_{SE}=\sum_{i\in S,,j\in E} J_{ij},\boldsymbol\sigma_i!\cdot!\boldsymbol\sigma_j). The coupling constants (J_{ij}) are varied to explore both weak and strong coupling regimes.

Time evolution is performed by directly integrating the time‑dependent Schrödinger equation for the full wavefunction (|\Psi(t)\rangle). After each time step the environment degrees of freedom are traced out to obtain the reduced density matrix (\rho_S(t)=\mathrm{Tr}_E|\Psi(t)\rangle\langle\Psi(t)|). The authors monitor the eigenvalues ({\lambda_k(t)}) of (\rho_S(t)) and the von Neumann entropy (S(t)=-\mathrm{Tr}