1. Discrete, Amorphous Physical Models Erik Rauch Digital Perspectives on Physics Workshop

2. Contribution It is possible to relax some of the basic assumptions of current discrete models and still model some important physical phenomena

3. Overview There are assumptions built in to all current discrete models, among others: –Regular lattice geometry –Synchronous updating Introduce new kind of discrete model which doesn’t have these two characteristics Show that such models can successfully model physical phenomena

4. Continuous physical models Maxwell’s equations (electromagnetics) Wave equation Navier-Stokes equation (fluid flow) Examples:

5. Discrete models Proposed as an alternative to, rather than approximation of, continuous models (Tom Toffoli) Examples are cellular automata, lattice gases –Successful in modelling: wave propagation, fluid dynamics, magnetization, etc. Agree with continuous models averaged over space and time Have been viewed as “first class” models: not an approximation of another model (e.g. Navier-Stokes equation)

6. Characteristics shared by discrete models: –Discrete space: space is a set of discrete sites, each with a state –Discrete time: change happens in discrete steps –Sites are arranged in a regular lattice –Global synchronization: all sites update simultaneously –Locality: sites are connected to nearest neighbors on lattice; new state of site is a function of states of connected sites Are all of these necessary? Is it possible to make a useful model that is more minimal in this sense? Motivation Physical modelling: strive to find a minimal description (Occam’s razor)

7. Motivation cont’d. Not required for a model that exhibits 'physical' behavior: –exhibits phenomena that are qualitatively like phenomena in physics –gives quantitatively similar results to conventional models of physical phenomena Regular lattice Global synchronization –updates do not happen independently, but are synchronized with all other sites

8. Motivation cont’d. Commonly used discrete models correspond to the regular structure of parallel computers Amorphous model: amorphous computer? –Bulk fabrication of cheap but unreliable processors –Connectivity is local, and not specified in advance –Limited or no global communication

9. In order to be physical, structure of irregular lattice should resemble real space Constraints are now statistical –Locality constraint: Small number of neighbors neighbors have a probability >> 0 of being neighbors of each other. –# of sites reachable in n hops ~ n D where D is dimension –A low variance in the neighborhood size –Regular lattice is a special case Relaxing the geometry assumption Not spacelike: Bethe lattice Spacelike

10. Sites are not synchronized with each other –Update independently –To resemble physical world, statistical constraint: low variance in the number of updates per site over time –To compare different levels of asynchrony, vary degree of randomness each pair updates at interval T + X; X is a random variable chosen on each update, << T Relaxing the synchronization assumption

11. Show that small change in lattice geometry and update order produces small difference in results To show that model doesn’t depend on synchronization and a particular lattice geometry, show that it is insensitive to –change in neighborhood structure –change in update order therefore, Relaxing the assumptions cont’d.

12. Conventional discrete models: new state depends on several neighbors –Would force all members of neighborhood synchronize with each other Avoid as much synchronization as possible: at most two sites synchronize with each other Only pairwise interactions

13. Example: waves State of a site is two variables: amplitude q, momentum p Physical laws are conservative, so explicitly build in conservation. For waves, conserve momentum: Packet of momentum proportional to difference in amplitude, as if sites were connected by springs (Hooke’s Law)

14. Quantitative comparison Compare by embedding lattice in Euclidean space and assigning each site coordinates Compute average difference of each lattice site's state with the continuous model's value at that point Difference grows linearly and slowly, as with conventional discrete models

15. Energy conservation Momentum-conserving model doesn’t conserve kinetic + potential energy difference in amplitude q i - q j between neighbors  potential energy momentum p  kinetic energy (directly related to a change in q) so energy cannot be described for a single site but only for a given pair Each site is potentially a member of many pairs, so changing q changes the potential energy in a way that depends on all neighbors simultaneously. –To conserve energy, do we need synchronization?

16. Energy- and momentum-conserving model l ij = potential energy associated with a bond use p i and l ij as state rather than p i and q i Can accomplish conservation of energy using ‘leapfrog’ update:

17. Energy- and momentum-conserving cont’d Solve for the new values in terms of the old: Determinant of transformation matrix T = 1, implying energy conservation

18. Conclusion Introduced alternate class of discrete models to show that 2 of the characteristics of conventional discrete models –crystalline geometry –global synchronization are not necessary for a useful model. Not very practical on current computer architectures, however: –A variant could be useful if highly parallel computers with corresponding structure become feasible (amorphous computers)