Financial Friction and Multiplicative Markov Market Game

We study long-term growth-optimal strategies on a simple market with linear proportional transaction costs. We show that several problems of this sort can be solved in closed form, and explicit the non-analytic dependance of optimal strategies and expected frictional losses of the friction parameter. We present one derivation in terms of invariant measures of drift-diffusion processes (Fokker- Planck approach), and one derivation using the Hamilton-Jacobi-Bellman equation of optimal control theory. We also show that a significant part of the results can be derived without computation by a kind of dimensional analysis. We comment on the extension of the method to other sources of uncertainty, and discuss what conclusions can be drawn about the growth-optimal criterion as such.Comment: 10 pages, invited talk at the European Physical Society conference 'Applications of Physics in Financial Analysis', Trinity College, Dublin, Ireland, July 14-17, 199

On the efficiency of heat engines at the micro-scale and below

We investigate the thermodynamic efficiency of sub-micro-scale heat engines operating under the conditions described by over-damped stochastic thermodynamics. We prove that at maximum power the efficiency obeys for constant isotropic mobility the universal law $\eta=2\,\eta_{C}/(4-\eta_{C})$ where $\eta_{C}$ is the efficiency of an ideal Carnot cycle. The corresponding power optimizing protocol is specified by the solution of an optimal mass transport problem. Such solution can be determined explicitly using well known Monge--Amp\`ere--Kantorovich reconstruction algorithms. Furthermore, we show that the same law describes the efficiency of heat engines operating at maximum work over short time periods. Finally, we illustrate the straightforward extension of these results to cases when the mobility is anisotropic and temperature dependent.Comment: 5 pages; revised version including the derivation of the efficiency and of the corresponding optimal protocols in the presence of anisotropic temperature dependent mobilit

An Application of Pontryagin’s Principle to Brownian Particle Engineered Equilibration

We present a stylized model of controlled equilibration of a small system in a fluctuating environment. We derive the optimal control equations steering in finite-time the system between two equilibrium states. The corresponding thermodynamic transition is optimal in the sense that it occurs at minimum entropy if the set of admissible controls is restricted by certain bounds on the time derivatives of the protocols. We apply our equations to the engineered equilibration of an optical trap considered in a recent proof of principle experiment. We also analyze an elementary model of nucleation previously considered by Landauer to discuss the thermodynamic cost of one bit of information erasure. We expect our model to be a useful benchmark for experiment design as it exhibits the same integrability properties of well-known models of optimal mass transport by a compressible velocity field

R. F\"urth's 1933 paper "On certain relations between classical Statistics and Quantum Mechanics" ["\"Uber einige Beziehungen zwischen klassischer Statistik und Quantenmechanik", \textit{Zeitschrift f\"ur Physik,} \textbf{81} 143-162]

We present a translation of the 1933 paper by R. F\"urth in which a profound analogy between quantum fluctuations and Brownian motion is pointed out. This paper opened in some sense the way to the stochastic methods of quantization developed almost 30 years later by Edward Nelson and others.Comment: 23 pages, 4 figure

Quantum trajectory framework for general time-local master equations

The paper was originally submitted as "Interference of Quantum Trajectories". We changed the title after a suggestion of the editorsMaster equations are one of the main avenues to study open quantum systems. When the master equation is of the Lindblad-Gorini-Kossakowski-Sudarshan form, its solution can be "unraveled in quantum trajectories" i.e., represented as an average over the realizations of a Markov process in the Hilbert space of the system. Quantum trajectories of this type are both an element of quantum measurement theory as well as a numerical tool for systems in large Hilbert spaces. We prove that general time-local and trace-preserving master equations also admit an unraveling in terms of a Markov process in the Hilbert space of the system. The crucial ingredient is to weigh averages by a probability pseudo-measure which we call the "influence martingale". The influence martingale satisfies a 1d stochastic differential equation enslaved to the ones governing the quantum trajectories. We thus extend the existing theory without increasing the computational complexity. Quantum trajectory frameworks describe systems weakly coupled to their environment. Here, by including an extra 1D variable in the dynamics, the authors introduce a quantum trajectory framework for time local master equations derived at strong coupling while keeping the computational complexity under control.Peer reviewe

Exactly solvable model of calorimetric measurements

Calorimetric measurements are experimentally realizable methods to assess thermodynamics relations in quantum devices. With this motivation in mind, we consider a resonant level coupled to a Fermion reservoir. We consider a transient process in which the interaction between the level and the reservoir is initially switched on and then switched off again. We find the time dependence of the energy of the reservoir, of the energy of the level, and of the interaction energy between them at weak, intermediate, strong, and ultrastrong coupling. We also determine the statistical distributions of these energies.Peer reviewe