Quantum emulation of classical dynamics

In statistical mechanics, it is well known that finite-state classical lattice models can be recast as quantum models, with distinct classical configurations identified with orthogonal basis states. This mapping makes classical statistical mechanics on a lattice a special case of quantum statistical mechanics, and classical combinatorial entropy a special case of quantum entropy. In a similar manner, finite-state classical dynamics can be recast as finite-energy quantum dynamics. This mapping translates continuous quantities, concepts and machinery of quantum mechanics into a simplified finite-state context in which they have a purely classical and combinatorial interpretation. For example, in this mapping quantum average energy becomes the classical update rate. Interpolation theory and communication theory help explain the truce achieved here between perfect classical determinism and quantum uncertainty, and between discrete and continuous dynamics.Comment: 8 pages, 1 figur

Crystalline Computation

A cellular automaton is a deterministic and exactly computable dynamical system which mimics certain fundamental aspects of physical dynamics such as spatial locality and finite entropy. CA systems can be constructed which have additional attributes that are basic to physics: systems which are exactly invertible at their finest scale, which obey exact conservation laws, which support the evolution of arbitrary complexity, etc. In this paper, we discuss techniques for bringing CA models closer to physics, and some of the interesting consequences of doing so.Comment: 39 pages, 15 figures (48 images), to appear in the book Feynman and Computation, A. Hey ed., (c) Perseus Books (1998

The maximum speed of dynamical evolution

We discuss the problem of counting the maximum number of distinct states that an isolated physical system can pass through in a given period of time---its maximum speed of dynamical evolution. Previous analyses have given bounds in terms of the standard deviation of the energy of the system; here we give a strict bound that depends only on E-E0, the system's average energy minus its ground state energy. We also discuss bounds on information processing rates implied by our bound on the speed of dynamical evolution. For example, adding one Joule of energy to a given computer can never increase its processing rate by more than about 3x10^33 operations per second.Comment: 14 pages, no figures, LaTex2e (elsart). This is the published version, which includes brief semi-classical and relativistic discussions not included in the original preprin

Universal Cellular Automata Based on the Collisions of Soft Spheres

Fredkin's Billiard Ball Model (BBM) is a continuous classical mechanical model of computation based on the elastic collisions of identical finite-diameter hard spheres. When the BBM is initialized appropriately, the sequence of states that appear at successive integer time-steps is equivalent to a discrete digital dynamics. Here we discuss some models of computation that are based on the elastic collisions of identical finite-diameter soft spheres: spheres which are very compressible and hence take an appreciable amount of time to bounce off each other. Because of this extended impact period, these Soft Sphere Models (SSM's) correspond directly to simple lattice gas automata--unlike the fast-impact BBM. Successive time-steps of an SSM lattice gas dynamics can be viewed as integer-time snapshots of a continuous physical dynamics with a finite-range soft-potential interaction. We present both 2D and 3D models of universal CA's of this type, and then discuss spatially-efficient computation using momentum conserving versions of these models (i.e., without fixed mirrors). Finally, we discuss the interpretation of these models as relativistic and as semi-classical systems, and extensions of these models motivated by these interpretations.Comment: 22 pages, 17 figures; published in 2002 as a chapter of a book (arxiv-ing for longterm availability

Counting distinct states in physical dynamics

A finite physical system has a finite entropy -- a finite number of distinct possible states. Here we show that finite maximum counts of distinct (orthogonal) states also define other basic quantities of physics, including energy, momentum and Lagrangian action. A finite number of distinct states also limits the resolution of measurements and makes classical spacetime effectively discrete. Our analysis generalizes speed limits on time evolution: we count the distinct states possible in a finite-length of unitary transformation. As in Nyquist's bound on distinct signal values in classical waves, widths of superpositions bound the distinct states per unit length. Maximally distinct transformations are effectively discrete, allowing us to simplify analysis and simulation of maximally distinct dynamics.Comment: 16 pages, 17 figures, ancillary files contain numerical test

Mechanical Systems that are both Classical and Quantum

Quantum dynamics can be regarded as a generalization of classical finite-state dynamics. This is a familiar viewpoint for workers in quantum computation, which encompasses classical computation as a special case. Here this viewpoint is extended to mechanics, where classical dynamics has traditionally been viewed as a macroscopic approximation of quantum behavior, not as a special case. When a classical dynamics is recast as a special case of quantum dynamics, the quantum description can be interpreted classically. For example, sometimes extra information is added to the classical state in order to construct the quantum description. This extra information is then eliminated by representing it in a superposition as if it were unknown information about a classical statistical ensemble. This usage of superposition leads to the appearance of Fermions in the quantum description of classical lattice-gas dynamics and turns continuous-space descriptions of finite-state systems into illustrations of classical sampling theory. A direct mapping of classical systems onto quantum systems also allows us to determine the minimum possible energy scale for a classical dynamics, based on a localized rate of state change. We use a partitioning description of dynamics to define locality, and discuss the ideal energy of two model systems.Comment: 24 pages, 10 figures. v2 fixed unclear wording and added one referenc

The finite-state character of physical dynamics

Finite physical systems have only a finite amount of distinct state. This finiteness is fundamental in statistical mechanics, where the maximum number of distinct states compatible with macroscopic constraints defines entropy. Here we show that finiteness of distinct state is similarly fundamental in ordinary mechanics: energy and momentum are defined by the maximum number of distinct states possible in a given time or distance. More generally, any moment of energy or momentum bounds distinct states in time or space. These results generalize both the Nyquist bandwidth-bound on distinct values in classical signals, and quantum uncertainty bounds. The new certainty bounds are achieved by finite-bandwidth evolutions in which time and space are effectively discrete, including quantum evolutions that are effectively classical. Since energy and momentum count distinct states, they are defined in finite-state dynamics, and they relate classical mechanics to finite-state evolution.Comment: 15 pages, 17 figures, ancillary file with mathematica noteboo

Finite-State Classical Mechanics

Reversible lattice dynamics embody basic features of physics that govern the time evolution of classical information. They have finite resolution in space and time, don't allow information to be erased, and easily accommodate other structural properties of microscopic physics, such as finite distinct state and locality of interaction. In an ideal quantum realization of a reversible lattice dynamics, finite classical rates of state-change at lattice sites determine average energies and momenta. This is very different than traditional continuous models of classical dynamics, where the number of distinct states is infinite, the rate of change between distinct states is infinite, and energies and momenta are not tied to rates of distinct state change. Here we discuss a family of classical mechanical models that have the informational and energetic realism of reversible lattice dynamics, while retaining the continuity and mathematical framework of classical mechanics. These models may help to clarify the informational foundations of mechanics.Comment: 14 pages, 4 figures, to be presented at 10th Conference on Reversible Computation, Leicester England, 13-14 September 201

A thermodynamically reversible generalization of Diffusion Limited Aggregation

We introduce a lattice gas model of cluster growth via the diffusive aggregation of particles in a closed system obeying a local, deterministic, microscopically reversible dynamics. This model roughly corresponds to placing the irreversible Diffusion Limited Aggregation model (DLA) in contact with a heat bath. Particles release latent heat when aggregating, while singly connected cluster members can absorb heat and evaporate. The heat bath is initially empty, hence we observe the flow of entropy from the aggregating gas of particles into the heat bath, which is being populated by diffusing heat tokens. Before the population of the heat bath stabilizes, the cluster morphology (quantified by the fractal dimension) is similar to a standard DLA cluster. The cluster then gradually anneals, becoming more tenuous, until reaching configurational equilibrium when the cluster morphology resembles a quenched branched random polymer. As the microscopic dynamics is invertible, we can reverse the evolution, observe the inverse flow of heat and entropy, and recover the initial condition. This simple system provides an explicit example of how macroscopic dissipation and self-organization can result from an underlying microscopically reversible dynamics.Comment: 13 pages, 8 figures, 1 table. Submitted to Phys. Rev.

Dimension-splitting for simplifying diffusion in lattice-gas models

We introduce a simplified technique for incorporating diffusive phenomena into lattice-gas molecular dynamics models. In this method, spatial interactions take place one dimension at a time, with a separate fractional timestep devoted to each dimension, and with all dimensions treated identically. We show that the model resulting from this technique is equivalent to the macroscopic diffusion equation in the appropriate limit. This technique saves computational resources and reduces the complexity of model design, programming, debugging, simulation and analysis. For example, a reaction-diffusion simulation can be designed and tested as a one-dimensional system, and then directly extended to two or more dimensions. We illustrate the use of this approach in constructing a microscopically reversible model of diffusion limited aggregation as well as in a model of growth of biological films.Comment: 13 pages, 5 figures, Latex2e, Revtex4. Added footnotes clarifying distinction between LGA and LB methods. Accepted to J Stat Phys, Nov 200