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

1. Discrete, Amorphous Physical ModelsErik RauchDigital 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 Examples: Maxwell’s equations (electromagnetics) Wave equation Navier-Stokes equation (fluid flow)

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. Motivation • Physical modelling: strive to find a minimal description (Occam’s razor) • 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?

7. Motivation cont’d. • Regular lattice • Global synchronization • updates do not happen independently, but are synchronized with all other sites • 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

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. Relaxing the geometry assumption • 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 ~ nD where D is dimension • A low variance in the neighborhood size • Regular lattice is a special case Not spacelike: Bethe lattice • Spacelike

10. Relaxing the synchronization assumption • 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, <|X|> << T

11. Relaxing the assumptions cont’d. • 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 • Show that small change in lattice geometry and update order produces small difference in results therefore,

12. Only pairwise interactions • 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

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) amplitude

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 qi - qj 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 • lij = potential energy associated with a bond • use pi and lij as state rather than pi and qi • 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)