Papers by Norman Margolus
We introduce a lattice gas model of cluster growth via the diffusive aggregation of particles in ... more 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.
Physical Review E, 1999
We introduce a lattice gas model of cluster growth via the diffusive aggregation of particles in ... more 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.
We introduce a simplified technique for incorporating diffusive phenomena into lattice-gas molecu... more 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.
The maximum computational density allowed by the laws of physics is available only in a format th... more The maximum computational density allowed by the laws of physics is available only in a format that mimics the basic spatial locality of physical law. Fine-grained uniform computations with this kind of local interconnectivity (Cellular Automata) are particularly good candidates for efficient and massive microphysical implementation.
Journal of Statistical Physics, 1995
We demonstrate how three-dimensional fluid flow simulations can be carried out on the Cellular Au... more We demonstrate how three-dimensional fluid flow simulations can be carried out on the Cellular Automata Machine 8 (CAM-8), a special-purpose computer for cellular automata computations. The principal algorithmic innovation is the use of a lattice gas model with a 16-bit collision operator that is specially adapted to the machine architecture. It is shown how the collision rules can be optimized to obtain a low viscosity of the fluid. Predictions of the viscosity based on a Boltzmann approximation agree well with measurements of the viscosity made on CAM-8. Several test simulations of flows in simple geometries—channels, pipes, and a cubic array of spheres-are carried out. Measurements of average flux in these geometries compare well with theoretical predictions.
Tommoso Toffoli Cellular Automata Machines A New Environment for Modeling ... Cellular Automata M... more Tommoso Toffoli Cellular Automata Machines A New Environment for Modeling ... Cellular Automata Machines This One 2QK4-UXQ-165T ... MIT Press Series in Scientific Computation Dennis Gannon, editor The Massively Parallel Processor, edited by JL Potter, 1985 Parallel ...
Physical Review Letters, 1986
Physica D-nonlinear Phenomena, 1990
In the light of recent developments in the theory of invertible cellular automata, we attempt to ... more In the light of recent developments in the theory of invertible cellular automata, we attempt to give a unified presentation of the subject and discuss its relevance to computer science and mathematical physics.
Physica D-nonlinear Phenomena, 1991
A new approach to polymer simulation well suited to massively parallel architectures is presented... more A new approach to polymer simulation well suited to massively parallel architectures is presented. The approach is based on a novel two-space algorithm that enables 50% of the monomers to be updated in parallel. The simplicity of this algorithm enables implementation and comparison of different platforms. Such comparisons are relevant to a wide variety of scientific applications. We tested this algorithm on three commercially available machines, the MP-1, KSR1, and CM-2; and on a prototype of the CAM-8 architecture. Among the commercial machines we found the MP-1 provided the best performance for highly-parallel fine-grained simulations. Effective utilization of the KSR1 was achieved with attention to synchronization requirements. The small (8 node) CAM-8 prototype, with a kind and cost of hardware comparable to an engineering workstation, achieved a performance within a factor of two of the MP-1 for our application.
We discuss cellular automata (CA) architectural considerations that led to thedesign of the cam-8... more We discuss cellular automata (CA) architectural considerations that led to thedesign of the cam-8 CA machine; describe some of the spatial modeling tasks that CA'shave been applied to on this machine; and discuss some of the interesting practical andtheoretical modeling challenges that remain.1 IntroductionFor the past 20 years, feature sizes of computing elements have decreased at a ratherconstant exponential rate.
Physical Review B, 1988
ABSTRACT We point out the advantages of computing the dynamics of Ising spin glasses and related ... more ABSTRACT We point out the advantages of computing the dynamics of Ising spin glasses and related (e.g., bond or site dilution) systems in zero external field by using bond-energy variables bij=sisjJij instead of the usual spin variables and coupling constants.
Physical Review A, 1995
We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exc... more We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Boolean values (x, y) to (x, x ⊕ y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2 n )) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two-and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations.
Annals of The New York Academy of Sciences, 1986
Discrete lattice systems have had a long and productive history in physics. Examples range from e... more Discrete lattice systems have had a long and productive history in physics. Examples range from exact theoretical models studied in statistical mechanics to approximate numerical treatments of continuum models. There has, however, been relatively little attention paid to exact lattice models which obey an invertible dynamics: from any state of the dynamical system you can infer the previous state. This kind of microscopic reversibility is an important property of all microscopic physical dynamics. Invertible lattice systems become even more physically realistic if we impose locality of interaction and exact conservation laws. In fact, some invertible and momentum conserving lattice dynamics-in which discrete particles hop between neighboring lattice sites at discrete times-accurately reproduce hydrodynamics in the macroscopic limit.
Physica D-nonlinear Phenomena, 1984
Reversible Cellular Automata are computer-models that embody discrete analogues of the classical-... more Reversible Cellular Automata are computer-models that embody discrete analogues of the classical-physics notions of space, time, locality, and microscopic reversibility. They are offered as a step towards models of computation that are closer to fundamental physics.
For a number of years, we have studied the large-scale fine- grained limit of cellular-logic-arra... more For a number of years, we have studied the large-scale fine- grained limit of cellular-logic-array calculations and computers -- with particular emphasis on applications to physical simulation. Perhaps the most relevant lessons of this work for the FPGA community have to do with the applicability of virtual-processor techniques to these logic-array computations -- and by extension to the design of FPGAs themselves. These techniques allow us to tradeoff speed against size, to balance resources devoted to data storage with those devoted to processing, and to time-share communication resources as we share processors. An application area of particular interest to the FPGA community is in logic emulation, where a virtual processing approach lets us maximize useful processor cycles by having processing hardware follow computational wavefronts through arrays of virtual logic ('temporal pipelining'). This technique is of direct relevance to FPGA design. Our virtual processor approach is embodied in our indefinitely scalable CAM-8 cellular automata (CA) machine. Personal-computer- scale prototypes, designed and built at MIT using 1988 technology, are still about as fast as any conventional computer for most large-scale physical CA applications. Using today's high-bandwidth DRAMs, machines with the same number of memory chips could be built that run 100 times faster. Rather than build a new dedicated CA machine processor, it is attractive to instead add appropriate DRAM I/O and data-buffering circuitry to an FPGA design, to create a general-purpose class of FPGAs optimized for large- scale virtual-processor applications.
Physical Review E, 1997
We generalize the hydrodynamic lattice gas model to include arbitrary numbers of particles moving... more We generalize the hydrodynamic lattice gas model to include arbitrary numbers of particles moving in each lattice direction. For this generalization we derive the equilibrium distribution function and the hydrodynamic equations, including the equation of state and the prefactor of the inertial term that arises from the breaking of galilean invariance in these models. We show that this prefactor can be set to unity in the generalized model, therby effectively restoring galilean invariance. Moreover, we derive an expression for the kinematic viscosity, and show that it tends to decrease with the maximum number of particles allowed in each direction, so that higher Reynolds numbers may be achieved. Finally, we derive expressions for the statistical noise and the Boltzmann entropy of these models.
... sites. Within the next few years a large CAM-8 sponsored by the US Air Force ... There are al... more ... sites. Within the next few years a large CAM-8 sponsored by the US Air Force ... There are also appealing reasons related to modeling physical systems with complex boundary conditions. The ... Quantum mechanics requires unitary, and hence invertible, time evolutionthe ...
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Papers by Norman Margolus