The Universe Is a ComputerEpisode 9 (extended) ／ The equations of information and the equations of physics

Taking seriously: "combine the equations of information, take the degrees of freedom and make it exact, and you get the differential equations of physics"

The equations of information and the equations of physics The equations of information theory count (finite, algebraic, integer). The equations of physics flow (continuous, differential).
The operation that joins them is just one ── take the degrees of freedom to \(\infty\) and demand exactness. Then counting turns into differentiation.

Core: information = finite, counts / physics = continuous, differential / bridge = dof→∞ Hook: equations of information \(\xrightarrow{\text{dof}\to\infty}\) differential equations

Last time we opened \(D\) to complex numbers and chased "is the cutoff real." This time, one step before that, a sturdier question ── what is the difference between the equations of information and the equations of physics? The answer is clean. The equations of information are a ledger (counting states); the equations of physics are a flow (tracking a rate). And the operation that turns a ledger into a flow is just one: take the degrees of freedom to infinity and make it exact. What's more, there is a concrete case where that literally spits out the Einstein equations, already inside your mass series.

01Two kinds of equations ── those that count, and those that flow

First, place them side by side. Just sorting the equations that came up over these few episodes left and right makes the difference visible.

Equations of information (count, finite, no derivatives)Equations of physics (flow, continuous, differential)
\(S=k\ln W\) (counting the number of states)Diffusion / heat equation \(\partial_t\rho=D\nabla^2\rho\)
\(S\le A/4\) (the ceiling on bits)Einstein equations \(G_{ab}+\Lambda g_{ab}=8\pi G\,T_{ab}\)
\(kT\ln2\) (the price of erasing one bit)Thermodynamic differential relation \(dU=T\,dS-p\,dV\)
\(F=1/(Cn)^D\) (the accounting of dimension)Wave / field equation \(\Box\,\phi=0\)
Transfer matrices / circuits (finite-dimensional linear algebra)Schrödinger equation \(i\hbar\,\partial_t\psi=H\psi\)

The left is integers and algebra, the right is derivatives. The left has not a single derivative; the right has derivatives everywhere. This apparent gap is, in fact, bridged by a single operation.

02The bridge is one ── a difference becomes a derivative

A difference chopped at lattice spacing \(\varepsilon\) (= the unit of resolution, a finite degree of freedom) becomes a derivative as \(\varepsilon\to0\):

The naive mechanism of ledger → flow $$\frac{f(x+\varepsilon)-f(x)}{\varepsilon}\ \xrightarrow[\ \varepsilon\to0\ ]{}\ \frac{df}{dx},\qquad \sum_i f(x_i)\,\varepsilon\ \xrightarrow[\ \varepsilon\to0\ ]{}\ \int f\,dx$$

A difference (subtraction = counting) becomes a derivative (a rate), and a sum (counting) becomes an integral. The moment you take the degrees of freedom \(N=L/\varepsilon\) to \(\infty\), an algebraic identity turns into a dynamical differential law. That \(S=k\ln W\) (counting) becomes, in the thermodynamic limit, smooth thermodynamics and diffusion equations (flow) is this same operation.

Figure 1: increase the degrees of freedom \(N\) and a difference (ledger) becomes a derivative (flow). At one point on the curve, the slope of the difference (terracotta secant) approaches the true derivative (teal tangent) as \(\varepsilon=L/N\to0\). The more points you add, the more the piecewise line (discrete version) clings to the curve. The error vanishes roughly in proportion to \(\varepsilon\).
Increase N with the slider and the slope of the difference approaches the true derivative.
true continuous curve f(x) secant of the difference (information = counting) true derivative = tangent (physics = flow)
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03The decisive example ── the Einstein equations fall out of the equations of information

This is not abstraction. There is a derivation where your single sentence literally comes true ── Jacobson 1995 (the one done in Appendix ⑥ of the mass series):

Equations of information + all degrees of freedom + exactness = differential equations

Put in the equations of information:

$$\underbrace{\delta S=\eta\,\delta A}_{\text{area entropy (counting)}}\quad \underbrace{T=\frac{\hbar\kappa}{2\pi}}_{\text{Unruh temperature}}\quad \underbrace{\delta Q=T\,\delta S}_{\text{Clausius}}$$

Demand that these hold exactly for every local Rindler horizon (= taking all directions and degrees of freedom), and:

$$\boxed{\,R_{ab}-\tfrac12 R\,g_{ab}+\Lambda g_{ab}=8\pi G\,T_{ab}\,}$$

Einstein's differential equations fall out. \(G\) is the coefficient of area entropy, and \(\Lambda\) an integration constant. Exactly "combine the equations of information, take the degrees of freedom, make it exact, and you get the differential equations of physics."

There are other examples with the same bones ── all of them "equations of information + exact over all degrees of freedom = differential equations":

Bridges of the same shape

04Restating the difference ── which is "the real thing"?

By now the difference condenses into a single arrow:

The relation between the equations of information and the equations of physics
$$\underbrace{\text{equations of information}}_{\text{finite, count, exact}}\ \xrightarrow[\ \text{demand exactness}\ ]{\ \text{degrees of freedom}\to\infty\ }\ \underbrace{\text{differential equations of physics}}_{\text{continuous, flow, idealized}}$$

A derivative is an operation premised on "you can zoom in infinitely." So the differential equations of physics are the story a finite, counting substrate tells about itself once it stops counting individual bits and follows only the average. The derivative is a fiction of the continuum limit.

Placed against last time's "the cutoff is real (the universe is a lazybones)," the ordering becomes clear. If the substrate really is finite ── the equations of information are the real thing (finite, exact), and the differential equations of physics are their \(\varepsilon\to0\) idealization = a convenient lie.

The reverse direction of last time's "cut off and compute" Episode 8's "let's cut off at some point and compute (effective field theory)" was the direction of reducing degrees of freedom. This time's "dof→∞ gives differential equations" is the direction of increasing them. The two are the two ends of one and the same axis ── in practice you cut off and compute (finite), and the beautiful continuous PDE of theory is the never-reached ideal of "this is what you'd get if you didn't cut off." In a lazy universe, that ideal image is never reached. So differential equations are always emergent / approximate, never fundamental ── that is the consistent reading of the "cutoff is real" camp.
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05The honest line ── the continuum limit isn't free

This bridge is real, but it rides on a one-way assumption. Three points, honestly:

(1) The continuum limit stands only at a critical point. A decent differential equation comes out at \(\varepsilon\to0\) only when the discrete theory sits at a critical point (an RG fixed point, correlation length \(\xi\to\infty\)). An ordinary finite, discrete theory produces lattice artifacts = Lorentz breaking (the very thing that disqualified the lattice CA in Episode 2). So "take the degrees of freedom and you get physics" is not unconditional.

Figure 2: the continuum limit stands only near a critical point. The vertical axis is correlation length \(\xi\) ÷ lattice spacing \(a\). Approach the critical point \(g_c\) and \(\xi/a\to\infty\) (the lattice spacing becomes negligible, so the continuum / differential equations stand and Lorentz is recovered). Move away and \(\xi/a\) is only a few ── the lattice is visible, and with artifacts = Lorentz breaking, the differential equations are not even an approximation. The slider \(g\) sets where you are now.
Bring g near the critical point g_c=0.5 with the slider and the continuum limit stands.
correlation length ξ/a continuum limit OK (ξ≫a) critical point g_c

(2) Jacobson assumes the equations of information to get differential equations. He takes the area law \(S=\eta A\) as input and derives the Einstein equations, but does not derive the area law itself. The mystery merely moves to "why \(S=A/4\)" = a microscopic state-count, and does not close (exactly the honest line of Appendix ⑥). The entanglement version and the Verlinde version have the same structure.

(3) The direction is reversed. This "dof→∞" is the opposite direction of last time's "cut off and compute." That the two are the two ends of the same axis is itself certain, but whether a "correct" continuum limit really exists from a given finite theory requires criticality, and in gravity it is unproven (the continuum limit of spin foams, the second-order phase transition of CDT). This is the the one hole the two series have consistently pointed at.

The honest line

"Equations of information → differential equations" is a real mechanism (difference→derivative, \(S=k\ln W\)→thermodynamics, Boltzmann→Navier–Stokes, transfer matrix→Schrödinger, and Jacobson→Einstein). There is no new physics here; all are established mathematics / physics.

However ── the ranking that "the equations of information are the real thing and differential equations are an emergent idealization" is not an established fact but your bet (only if the cutoff is real). The physics side can equally be read the other way, "the continuum is real and the discrete is an approximation" (Episode 3: continuous / discrete are representations, directions of reading). Which is fundamental hangs on the fact that the continuum limit stands only at criticality, and on the microscopic origin of \(S=A/4\) = the unsolved hole. "Do not say solved" ── the bridge was built; which bank is the foundation is left undecided.

Practice problems (solvable from this sheet)
  1. State in one phrase the apparent difference between the equations of information and the equations of physics. And what is the operation that joins them?
    See the answer
    The equations of information "count" (finite, algebraic, integer, no derivatives); the equations of physics "flow" (continuous, differential). The joining operation is one ── take the degrees of freedom to \(\infty\) and demand exactness (\(\varepsilon\to0\)). A difference becomes a derivative, a sum becomes an integral, and \(S=k\ln W\) becomes smooth thermodynamics.
  2. What is the most vivid example of "differential equations coming out of the equations of information"? And what is not derived there?
    See the answer
    Jacobson 1995: impose area entropy \(\delta S=\eta\delta A\) + Unruh temperature + Clausius \(\delta Q=T\delta S\) exactly for every local Rindler horizon, and the Einstein equations fall out. But the area law \(S=A/4\) itself is an assumption and is not derived. The mystery merely moves to "why \(S=A/4\)" = a microscopic state-count.
  3. Is "the equations of information are real and differential equations are an emergent idealization" a fact?
    See the answer
    Not a fact but a bet. It holds in the case of "the cutoff is real (the substrate is finite)." Conversely it can be read "the continuum is real and the discrete is an approximate representation" (Episode 3). The verdict hangs on the continuum limit standing only at a critical point, and on the microscopic origin of \(S=A/4\) = unsolved. The bridge (counting→differentiation) is established; which bank is the foundation is undecided.

SummaryCounting becomes differentiation once you take the degrees of freedom

The equations of information count (\(S=k\ln W\), \(S\le A/4\), \(kT\ln2\), \(F=1/(Cn)^D\); finite, algebraic, no derivatives); the equations of physics flow (Einstein, Schrödinger, Navier–Stokes; continuous, differential). What joins them is a single operation = take the degrees of freedom to \(\infty\) and make it exact. A difference becomes a derivative, a sum becomes an integral. And literally ── impose area entropy + Unruh + Clausius exactly on all horizons and the Einstein equations fall out (Jacobson). The first law of entanglement → linearized Einstein, Boltzmann → Navier–Stokes, transfer matrix → Schrödinger have the same bones.

In the "cutoff is real" reading, the equations of information are real and differential equations are the \(\varepsilon\to0\) idealization (= the reverse of last time's "cut off and compute"). But this ranking is a bet, not a fact ── the continuum limit stands only at a critical point (otherwise lattice artifacts = Lorentz breaking), Jacobson assumes \(S=A/4\), and its microscopic origin = the one hole stays open. The bridge was built; which bank is the foundation we honestly leave undecided.

← Episode 8: The imaginary dimension To the table of contents Episode 10: The dimensions of the four forces →
This document is Episode 9 (extended, a summary of a conversation, honest version) of the series "The Universe Is a Computer." The continuum limit of finite differences (difference→derivative, Riemann sum→integral), the limit from statistical mechanics to hydrodynamics (Boltzmann equation→Navier–Stokes), the continuous-time limit of the path integral / transfer matrix (→Schrödinger equation), thermodynamic gravity (Jacobson 1995: local Rindler horizons + area entropy + Unruh temperature + Clausius → Einstein equations), the linearized Einstein equations from the first law of entanglement (Van Raamsdonk, Faulkner, Lashkari et al.), the continuum limit and RG fixed points / critical points (divergent correlation length), and the lattice continuum-limit problem and recovery of Lorentz invariance (the continuum limit of spin foams, the second-order phase transition of CDT) are all established mathematics / physics, or current research topics. \(F=1/(Cn)^D\) is the defining relation of dimension; \(S\le A/4\), the CKN bound, and \(\hbar H_0/c^2\) are content honestly delineated in the sister series "Mass That Clicks." The ranking that "the equations of information are fundamental and differential equations are emergent" is not an established fact and depends on the unsolved bet of whether the substrate is finite (type I). Jacobson's derivation uses the area entropy law as an assumption, and its microscopic origin is not derived. ── To print, use your browser's "Print" and "Save as PDF" (in the print version the sliders and answers are static and hidden).

Print / PDF: Ctrl+P (⌘+P on Mac). On screen, in Figure 1 increasing the degrees of freedom N brings the difference toward the derivative, and in Figure 2 approaching the critical point makes the continuum limit stand. Click "See the answer" to reveal each solution.