Emergence of ordinal time

2 Emergence of ordinal time

Time is the ordinal index of a random walk through the most compressible configurations. The minimal—and hence most compressible—path is the path of least change between consecutive configurations. That path is realised by a fair random bit-flip mutation of an $n$ -bit string: at each discrete tick one of the $n$ bits is chosen uniformly and flipped.

This dynamics is the Ehrenfest process. Let $m$ be the number of 1-bits and $p = m ∕n$ . The expected change in one tick is

𝔼 [Δm ] = n-−-m − m- = 1− 2p. n n

(1)

In the continuum limit one obtains the ordinary differential equation

dp-= 1-−-2p, dt n

(2)

where $t$ is the raw flip count (“bit-flip time”). Starting from the all-zero state $p(0) = 0$ the solution is

1-( −2t∕n) p(t) = 2 1− e .

(3)

The relaxation timescale is therefore $O (n)$ ticks. Rescaling to the dimensionless coordinate $τ = t∕n$ yields the universal curve

1( −2τ) p(τ) = 2 1 − e , τ ∈ [0,∞ ).

(4)

The associated Shannon entropy per bit is

S(τ) ( ) -n---= H2 p(τ) ,

(5)

where $H (q) = − qlog q − (1− q)log (1− q) 2 2 2$ .

A further change of variable

x(τ) = 1− e−2τ ∈ [0,1)

(6)

produces a bounded time coordinate whose entropy is simply

S(x) ( ) -----= H2 x∕2 . n

(7)

This is the universal entropy curve that any physical time scale is ultimately mapped onto.

Compressibility is measured by spectral complexity: the number of bits required to encode the modes (frequencies and phases) of a wavefunction. The Ehrenfest trajectory is the path of least spectral complexity; it is therefore the typical path under the natural measure on configuration space. Solomonoff induction, when formulated with spectral rather than Kolmogorov complexity, preferentially samples exactly this trajectory.

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