In particular, we want to draw the attention of readers to the analogy between the probabilistic mass transport problem discovered by Schrödinger and the optimal control problem associated with the derivation of Landauer’s bound in non-equilibrium statistical nano-physics. Theoretical analysis then requires a careful overhaul of the meaning of time reversal and dissipation in systems whose evolution laws can only be defined in a statistical sense see, e.g. Fundamental questions concerning non-equilibrium statistical laws can be now posed in well-controlled experimental setups. The advent of nano-manipulations has made possible accurate laboratory observations of systems, natural and artificial, operating in contact with highly fluctuating environments. Our interest in presenting an English translation, so far missing to the best of our knowledge, of “On the Reversal of the Laws of Nature” is more directly motivated by the second part of the paper, especially § 6, where Schrödinger discusses fluctuations and time reversal in classical statistical physics. Kolmogorov’s 1937 paper on detailed balance has the title “On the Reversibility of the Statistical Laws of Nature” thus clearly resonating the title of Schrödinger’s paper.
Andreǐ Kolmogorov starts his 1936 “On the Theory of Markov Chains” paper discussing the interest of studying time reversal of Markov processes for the “ analysis of the reversibility of the statistical laws of nature” explicitly referring to Schrödinger’s paper. Sergei Bernstein discussed the contents of Schrödinger’s paper in his address to the International Conference of Mathematicians held in Zürich in 1932. įrom the mathematical angle, and in particular the theory of Markov processes, the importance of the ideas put forward by Schrödinger was immediately realized. In a spirit perhaps closer to Schrödinger’s original idea, “Euclidean Quantum Mechanics” applies modern developments of stochastic calculus of variations and optimal control theory (see for recent surveys) to study how “ relations between quantum physics and classical probability theory” may “ lead to new theorems in regular quantum mechanics”. The collection offers a recent appraisal of the state of the art focusing on Nelson’s contributions. A similar program was also pursued by Masao Nagasawa and Robert Aebi. After the second world war, Fürth’s work became the starting point of the “stochastic mechanics” program proposed by Imre Fényes and Edward Nelson as a probabilistic interpretation of quantum mechanics. Already in 1933, Reinhold Fürth showed (see for a translation) what in current mathematical language could be phrased as the existence of uncertainty relations satisfied by the variance of a martingale of a diffusion process times the variance of the martingale’s “current velocity”. The observation of this analogy initiated parallel, and somewhat intertwined, lines of research aiming at either finding classical probabilistic analogues of quantum mechanics or more directly a classical probabilistic interpretation of quantum mechanics. As Schrödinger himself stated in § 3 of the paper, his “ current concern” was to point out the existence of a classical probabilistic structure such that the probability density is given by “ the product of a certain solution of” “ and a certain solution of” and thus presenting “ a striking analogy with quantum mechanics” (§ 4).
The intellectual context of the paper was marked by the intense debate on the interpretation of quantum mechanics. Schrödinger presented “On the Reversal of the Laws of Nature” to the Academy in March 1931. During their common years in Berlin, Einstein and Schrödinger became good personal friends. Among physicists, other members of the Academy were Max Planck, who proposed Schrödinger’s membership, Max von Laue, Walther Nernst and Albert Einstein who held a special Academy professorship. At the moment of his election, at the age of forty-two, Schrödinger was the youngest member of the Academy. Erwin Schrödinger had the rare privilege to be elected to the Prussian Academy of Science in February 1929, about 1 year and a half after his appointment to the chair of theoretical physics at the University of Berlin.
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