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Brownian motion in a field of force and the diffusion model of chemical reactions

For centuries knowledge meant proven knowledge — proven either by the power of the intellect or by the evidence of the senses. Wisdom and intellectual integrity demanded that one must desist from unproven utterances and minimize, even in thought, the gap between speculation and established knowledge. The proving power of the intellect or the senses was questioned by the sceptics more than two thousand years ago; but they were browbeaten into confusion by the glory of Newtonian physics. Einstein’s results again turned the tables and now very few philosophers or scientists still think that scientific knowledge is, or can be, proven knowledge. But few realize that with this the whole classical structure of intellectual values falls in ruins and has to be replaced: one cannot simply water down the ideal of proven truth - as some logical empiricists do — to the ideal of’probable truth’1 or — as some sociologists of knowledge do — to ‘truth by [changing] consensus’.2

Falsification and the Methodology of Scientific Research Programmes

We adapt the ideas underlying the success of Deep Q-Learning to the continuous action domain. We present an actor-critic, model-free algorithm based on the deterministic policy gradient that can operate over continuous action spaces. Using the same learning algorithm, network architecture and hyper-parameters, our algorithm robustly solves more than 20 simulated physics tasks, including classic problems such as cartpole swing-up, dexterous manipulation, legged locomotion and car driving. Our algorithm is able to find policies whose performance is competitive with those found by a planning algorithm with full access to the dynamics of the domain and its derivatives. We further demonstrate that for many of the tasks the algorithm can learn policies end-to-end: directly from raw pixel inputs.

Continuous control with deep reinforcement learning

The linear response theory has given a general proof of the fluctuation-dissipation theorem which states that the linear response of a given system to an external perturbation is expressed in terms of fluctuation properties of the system in thermal equilibrium. This theorem may be represented by a stochastic equation describing the fluctuation, which is a generalization of the familiar Langevin equation in the classical theory of Brownian motion. In this generalized equation the friction force becomes retarded or frequency-dependent and the random force is no more white. They are related to each other by a generalized Nyquist theorem which is in fact another expression of the fluctuation-dissipation theorem. This point of view can be applied to a wide class of irreversible process including collective modes in many-particle systems as has already been shown by Mori. As an illustrative example, the density response problem is briefly discussed.

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The fluctuation-dissipation theorem

This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology "Dynamical Systems in Neuroscience" presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties The book introduces dynamical systems starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems Each chapter proceeds from the simple to the complex, and provides sample problems at the end The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum - or taught by math or physics department in a way that is suitable for students of biology This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience

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Dynamical Systems in Neuroscience

With a method first indicated by Ornstein the mean values of all the powers of the velocity $u$ and the displacement $s$ of a free particle in Brownian motion are calculated It is shown that $u\ensuremath{-}{u}_{0}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}t)$ and $s\ensuremath{-}\frac{{u}_{0}}{\ensuremath{\beta}[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}t)]}$ where ${u}_{0}$ is the initial velocity and $\ensuremath{\beta}$ the friction coefficient divided by the mass of the particle, follow the normal Gaussian distribution law For $s$ this gives the exact frequency distribution corresponding to the exact formula for ${s}^{2}$ of Ornstein and F\"urth Discussion is given of the connection with the Fokker-Planck partial differential equation By the same method exact expressions are obtained for the square of the deviation of a harmonically bound particle in Brownian motion as a function of the time and the initial deviation Here the periodic, aperiodic and overdamped cases have to be treated separately In the last case, when $\ensuremath{\beta}$ is much larger than the frequency and for values of $t\ensuremath{\gg}{\ensuremath{\beta}}^{\ensuremath{-}1}$, the formula takes the form of that previously given by Smoluchowski

On the Theory of the Brownian Motion

It is investigated whether suitable generalizations of the field equations of current field theories to equations of higher order may be of help in eliminating the divergent features of the present theory. It turns out to be difficult, if feasible, to reconcile in this way the requirements of convergence, of positive definiteness of the free field energy, and of a strictly causal behavior of the state vector of a physical system. Progress may perhaps be made by relinquishing the condition of unlimited localizability of any space time event.

On Field Theories with Non-Localized Action

In this paper the general theory of transport phenomena in simple gases is concluded, and numerical values of the gas coefficients for heat conductivity and viscosity are obtained as a function of the temperature and density for the particular case of molecules acting as rigid elastic spheres. In the Introduction will be found a qualitative discussion of the principal results obtained and an interpretation of these results according to elementary considerations. In Section 1 the method of solution of the integral equations with which the formal theory of Part I was concluded is given together with the resulting general equations for the heat conductivity and viscosity coefficients. Since the integral equations can be solved only by a method of successive approximations, the expressions for the gas coefficients are in the form of infinite series the rapidity of convergence of which depends on a suitable choice of a complete set of auxiliary functions. In Section 2 all of the integrals appearing in the first two terms of the infinite series are evaluated. That only two terms are required is due to the fact that one is able to make an excellent choice of functions to represent the auxiliary set. In the evaluation of these integrals restriction is made in the application of the theory to small values of the degeneracy parameter $A$, since only terms in the zeroth and first power of $A$ are retained. This restriction is only slightly greater than that imposed by the fundamental postulates of the general theory which restrict its applicability to moderately rare gases. In Section 3 the final equations for the gas coefficients are applied to gases consisting of molecules which interact quantum-mechanically as rigid elastic spheres of diameters and masses associated with the gases helium and hydrogen.

Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I

With the spinor analysis developed by B. van der Waerden which comprises all representations of the Lorentz group, even those not contained in ordinary tensor calculus, one is able to write all derivations and equations in an automatically covariant form. For the convenient translation into spinor language of the Maxwell equations, it becomes important to introduce three self-dual tensors, one representing the electromagnetic field, one corresponding to the Hertzian vector, and one representing a kind of current potential. These correspond to symmetric spinors of the 2nd rank. Many spinor equations thus become simpler than the corresponding tensorial equations, especially the expression for the stress energy tensor. From the 1st order Dirac equations in spinor form, as given by v.d. Waerden, we derived the 2nd order equation, which agrees with the Gordon-Klein form but for a correction term which again contains the self-dual field tensor. Further the expression for the current was derived, and its decomposition into conduction and polarization currents, and both Maxwell and Dirac equations were derived from a spinorial variation principle, analogous to the results of Gordon and Darwin. In addition to the divergence condition for the current three new invariant relations between the wave functions which are independent of the potentials were obtained (Chapter III, Eqs. (11), (12) and (13)).

G. E. Uhlenbeck

Papers

On the Theory of the Brownian Motion

On Field Theories with Non-Localized Action

Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I

Application of Spinor Analysis to the Maxwell and Dirac Equations

The Two-Minima Problem and the Ammonia Molecule