Rigorous results in nonequilibrium dynamics
Non-equilibrium statistical mechanics is still far from being fully understood. Under these circumstances, it is important to find rigorous laws that hold in nonequilibrium systems.
・Review on bounds in nonequilibrium dynamics
Understanding non-equilibrium systems is an challenging problem in physics. In recent years, the importance of the "inequalities" that govern non-equilibrium physics has been increasingly understood in recent years.
In this review, we survey various inequalities in such nonequilibrium systems (especially quantum systems), from the basics to the latest results. We discuss speed limits, quantum thermalization and equilibration, the Lieb-Robinson bounds, entanglement generation error bounds in approximation, and other miscellaneous topics.
International Journal of Modern Physics B [arXiv:2202.02011]
・Speed limits for macroscopic systems
Mandelstam and Tamm showed in 1945 that the speed of transition of a quantum state through unitary time evolution can be suppressed by using the energy fluctuations of the system. Since then, the speed limits have been actively studied in both quantum and classical systems with various applications such as quantum control. On the other hand, most conventional speed limits cannot be directly used for processes involving macroscopic transitions. For example, if we consider a situation where a particle is transported from one place to another, the Mandelstam-Tamm limit diverges and does not give us a tight inequality.
In this work, we have developed a general framework for deriving useful speed limits for macroscopic transitions on the basis of the continuity equation of probability. Considering a general graph, we show that the speed of a physical quantity defined on the graph can be upper bounded using the "smoothness" of the quantity and the local probability current. In particular, in the case of unitary quantum systems, we find that the speed limit can become smaller as the expectation value of the transition Hamiltonian increases (due to the suppression of quantum phase differences). Furthermore, we showed that a similar speed limit can be obtained for macroscopic quantum coherence. We also showed that this framework can be applied to stochastic processes with classical and quantum macroscopic transitions.
・Universal error bound for constrained dynamics
When a quantum system has a large enough gap between different energy scales, the effective physics can be approximated by a low-energy theory. This effective theory can also approximate the dynamics for short times, the approximation may become worse over time since approximation errors accumulate. We have rigorously established a universal upper bound on such an error in constrained dynamics. This is the first result that mathematically guarantees the validity of low-energy approximation in constrained dynamics (up to a certain time scale).
・Universal constraints in nonlinear population dynamics
Nonlinear systems exhibit a wide variety of dynamics, including chaos and bifurcation, and to find general laws in such systems is an important problem. We have focused on nonlinear systems in evolutionary and ecological models and derived an upper bound (via Fisher information) on the (appropriately defined) speed of physical quantities. This upper bound gives a generalization of Fisher's fundamental theorem, which is well known in the theory of evolution.
We also find that, near the bifurcation point, the speed and its information-theoretical upper bound exhibit universal dynamical scaling. We argue that the exponents associated with this scaling law for the speed have universal bounds determined only from the type of bifurcation. For example, we find that the lower bound for which the speed decays should be 1/2 for transcritical bifurcations and 3/2 for Hopf bifurcations, irrespective of the details of the system. Our general results have been confirmed in evolutionary models, SIR models, Lotka-Volterra models, etc.