This is an exploration with today’s tools of probability theory into the mystery of ‘Über die Umkehrung der Naturgesetze’ (On time-reversal of natural laws) written by Erwin Schrödinger in 1931. He propagates the idea that diffusions given their marginal distributions at finite initial and final time are time-reversible. We are going to meet clouds of identical and independent particles considering them as realizations of diffusions. The particle dynamics is assumed to be known. Given the observation of such a cloud in terms of distribution densities at finite initial and final time, the intention is to find the ‘most probable’ distribution density of the cloud at intermediate times.
This is an exploration with today’s tools of probability theory into the mystery of ‘Über die Umkehrung der Naturgesetze’ (On time-reversal of natural laws) written by Erwin Schrödinger in 1931. He propagates the idea that diffusions given their marginal distributions at finite initial and final time are time-reversible. We are going to meet clouds of identical and independent particles considering them as realizations of diffusions. The particle dynamics is assumed to be known. Given the observation of such a cloud in terms of distribution densities at finite initial and final time, the intention is to find the ‘most probable’ distribution density of the cloud at intermediate times.
Abstract A decade before his 1926 paperb reporting his discovery of wave mechanics, a derivation of the time-independent wave equation, and its solutions for hydrogen-like atoms, Erwin Schrödinger had carried out several studies dealing with Brownian motion. It is clear from several of the papers he wrote that he was completely familiar with diffusion equations, the Fokker-Planck equation in many forms, their connections with Green’s functions, and with behavior of systems of particles undergoing random walks including those in force fields.c Thus, it is not surprising that Schrödinger was aware of the relation of the Schrödinger equation in imaginary time, to the diffusion equation governing particles undergoing diffusion, In this paper, with the title “On the Reversal of Natural Laws,” Schrödinger pointed out the remarkable formal analogy of diffusion with quantum mechanics and the discrepancies introduced by the factor J=T. In his discussion Schrödinger made the statement, “I am unable to foresee whether this analogy will prove itself useful in the clarification of quantum mechanical concepts,” but he finished the paper with the hope it would. There is no indication that Schrödinger was aware of the possibilities of a next step into quantum Monte Carlo - the use of random walks to provide solutions to the wave equation. This was left for others to discover later.
Abstract We present an English translation of Erwin Schrödinger’s paper on “On the Reversal of the Laws of Nature‘’. In this paper, Schrödinger analyses the idea of time reversal of a diffusion process. Schrödinger’s paper acted as a prominent source of inspiration for the works of Bernstein on reciprocal processes and of Kolmogorov on time reversal properties of Markov processes and detailed balance. The ideas outlined by Schrödinger also inspired the development of probabilistic interpretations of quantum mechanics by Fényes, Nelson and others as well as the notion of “Euclidean Quantum Mechanics” as probabilistic analogue of quantization. In the second part of the paper, Schrödinger discusses the relation between time reversal and statistical laws of physics. We emphasize in our commentary the relevance of Schrödinger’s intuitions for contemporary developments in statistical nano-physics.
Our account begins in 1931/1932, when one of the fathers of quantum theory, Erwin Schrodinger, proposed a Gedankenexperiment to comprehend the propagation of a cloud of Brownian particles. He proposed [1], [2] the inference problem to identify the most likely evolution of a large collection of particles as they traverse between two endpoints in time- their distribution being measured and available at the two endpoints. This random evolution, now known as the Schrodinger bridge, entails a flow of one-time distributions that interpolate the known initial and final marginals.
The first question to ask about the phenomenon of relaxation is why it occurs at all. Both the Newton and the Schrödinger equations are symmetrical under time reversal: The Newton equation, dx/dt = v ; dv/dt = −∂V/∂x, implies that particles obeying this law of motion will retrace their trajectory back in time after changing the sign of both the time t and the particle velocities v. The Schrödinger equation, ∂ψ/∂t = −(i/.h) Ĥ ψ, implies that if (ψ (t) is a solution then ψ *(−t) is also one, so that observables which depend on |ψ|2 are symmetric in time. On the other hand, nature clearly evolves asymmetrically as asserted by the second law of thermodynamics. How does this asymmetry arise in a system that obeys temporal symmetry in its time evolution? Readers with background in thermodynamics and statistical mechanics have encountered the intuitive answer: Irreversibility in a system with many degrees of freedom is essentially a manifestation of the system “getting lost in phase space”:Asystem starts from a given state and evolves in time. If the number of accessible states is huge, the probability that the system will find its way back to the initial state in finite time is vanishingly small, so that an observer who monitors properties associated with the initial state will see an irreversible evolution. The question is how is this irreversible behavior manifested through the reversible equations of motion, and how does it show in the quantitative description of the time evolution. This chapter provides an introduction to this subject using the time-dependent Schrödinger equation as a starting point. Chapter 10 discusses more advanced aspects of this problem within the framework of the quantum Liouville equation and the density operator formalism.
The theory of Schr¨ odinger bridges for diffusion processes is extended to discrete-time Markov chains, and to some problems for quantum discrete-time processes. Taking into account the past-future lack of symmetry of the discrete-time setting, results bear a striking resemblance to the classical ones. In particular, the solution of the path space maximum entropy problems is always obtained from the model by means of a suitable multiplicative functional transformation. Index Terms—Markov chain, maximum entropy problem, Schr¨ odinger bridge, time reversal evolution, space-time har- monic function, quantum operation. odinger considered, and basically solved, the following abstract probabilistic problem: Suppose a large number N of independent Brownian particles have been observed to have density ρ0(x) at time t0 and density ρ1(x) at some later time t1. Suppose the latter density considerably differs from what is predicted by the law of large numbers. It is apparent that the particles have been transported in an unlikely way. But of the many unlikely ways in which this could have happened, which one is the most likely? In modern terminology, this is a problem of large deviations of the empirical distribution. Using a coarse graining approach, Schrcomputed the most likely endpoint distribution under the prior tran- sition density of the Brownian motion p(s,x,t,y) and with the prescribed marginals. It turned out that the solution, namely the bridge from ρ0 to ρ1 over Brownian motion, has at each time a density q that factors as q(x,t )= ϕ(x,t)ˆ ϕ(x,t), where ϕ and ˆ ϕ are, in the language of Doob, a p-harmonic and a p-coharmonic functions, respectively. The existence and uniqueness of such a pair (ϕ, ˆ ϕ) satisfying the factorization above and the boundary conditions was guessed by Schron the basis of his intuition. He was later shown to be quite right in various degrees of generality by Fortet (9), Beurling (4), Jamison (11), F¨