Geometric approach to optimal nonequilibrium control: Minimizing dissipation in nanomagnetic spin systems

Abstract

Optimal control of nanomagnets has become an urgent problem for the field of spintronics as technological tools approach thermodynamically determined limits of efficiency. In complex, fluctuating systems, such as nanomagnetic bits, finding optimal protocols is challenging, requiring detailed information about the dynamical fluctuations of the controlled system. We provide a physically transparent derivation of a metric tensor for which the length of a protocol is proportional to its dissipation. This perspective simplifies nonequilibrium optimization problems by recasting them in a geometric language. We then describe a numerical method, an instance of geometric minimum action methods, that enables computation of geodesics even when the number of control parameters is large. We apply these methods to two models of nanomagnetic bits: a Landau-Lifshitz-Gilbert description of a single magnetic spin controlled by two orthogonal magnetic fields, and a two-dimensional Ising model in which the field is spatially controlled. These calculations reveal nontrivial protocols for bit erasure and reversal, providing important, experimentally testable predictions for ultra-low-power computing.

A mean field view of the landscape of two-layer neural networks

“…Let us finally mention that, after a first version of this paper appeared as a preprint, several other groups obtained results that are closely related to Theorem 3 ( 25 – 27 ).…”

Theory of Nonequilibrium Free Energy Transduction by Molecular Machines

“…[231][232][233] Several groups have used this framework to examine optimal protocols in model systems. 148,[234][235][236][237][238][239][240] Applying this theory to bistable systems representing thermally activated processes 241 leads to the intuition that energetically efficient control requires relatively slow perturbation when the system is on the verge of a major transition, essentially letting random thermal fluctuations kick the system over a given barrier 'for free' without energy input from the controller. Other extensions have generalized this control framework to nonequilibrium steady states 242,243 and to models of rotary machines 244 and chemical reaction networks.…”

Geometric Optimisation of Quantum Thermodynamic Processes

“…The main ideas were introduced for classical systems in a series of seminal papers in the 80 s by Weinhold and Andresen, Berry and Salamon, among others [ 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 ]. More recently, the field saw a revival following a series of papers initiated by Crooks in 2007 [ 15 , 16 , 17 ], leading to several applications in, e.g., molecular motors [ 18 ], small-scale information processing [ 19 ], nonequilibrium steady states [ 20 , 21 ], and many-body systems [ 22 , 23 ]. The same ideas have been generalised to the quantum regime for unitary dynamics using linear response [ 24 , 25 , 26 , 27 , 28 ], and to open system dynamics for Lindbladian systems [ 29 , 30 ].…”