Finite-State Classical Mechanics

Newton C. A. da Costa, Francisco Antônio Dória

Norman Margolus

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.

Norman Margolus

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.

Peter Bongaarts

Norman Margolus

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.