Expansion, Equilibrium, and Macroscopic Closure#
Suppose a swarm has settled into a reproducible statistical state. You can measure its mean energy, follow the growth of a region, and ask how much of its future can be predicted from a few coarse variables. These are three different experiments, each with a precise mathematical description.
This chapter connects them through observable bounds, geometric evolution equations, and coarse-graining criteria. The probability results control relaxation to a QSD. Raychaudhuri’s equation controls the expansion of a specified congruence. Closure conditions describe when a reduced state carries enough information to predict its own future. A physical cosmology joins these pieces through a metric, a stress tensor, a constitutive equation, and calibrated units.
Boundary response, stationary statistics, and bulk geometry#
Start by labeling the measuring instruments. A boundary experiment measures how energy changes when an area changes. A statistical experiment averages an observable over surviving runs. A curvature experiment measures a metric. The first two definitions make the boundary and statistical quantities explicit, so we can see exactly what a later geometric identification must supply.
Definition 791 (Boundary energy response)
For a specified boundary energy \(E_\partial(A)\), define the surface response \(\Pi_\partial=-dE_\partial/dA\). Converting this quantity to a bulk energy density requires a length scale and a constitutive prescription. The Gaussian dilation derivative and correlation stiffness are computed in Theorem 443; their sign and units are retained in any such conversion. A surface response is not itself a cosmological constant.
Definition 792 (Stationary bulk reference)
Let \(\pi_N\) be the finite-particle QSD of the killed update established under Theorem 209. For an integrable energy observable \(E_N\), define its stationary reference \(e_N=\pi_N(E_N)\) and centered observable \(\widetilde E_N=E_N-e_N\). This centering specifies the energy zero for statistical comparisons. It does not specify an Einstein constant.
Theorem 458 (Centered observables at the QSD)
For the preceding reference and an initial law \(\pi_N\),
at every time with positive survival probability. Adding a constant to the energy observable changes \(e_N\) by the same constant and leaves this identity unchanged.
Proof
The defining QSD identity gives \(\mathcal L_{\pi_N}(X_t\mid\tau_\dagger>t)=\pi_N\). Integrate \(\widetilde E_N\) against this law. The result is \(\pi_N(E_N)-e_N=0\). Centering \(E_N+C\) gives the same centered observable. No geometric field equation enters this argument.
Centering an observable is like choosing the zero on a measuring scale. Once you subtract its stationary mean, its mean at stationarity is zero by construction. Every individual configuration can still fluctuate, and the conditioned law can stay fixed while fewer runs survive. The theorem uses precisely this conditioned-law statement.
The next quantity has a measurable relaxation rate: the difference between the current mean and its stationary value. A bound on probability laws becomes a bound on that difference by testing the laws against the chosen observable.
Definition 793 (Nonequilibrium observable excess)
For a conditioned law \(\mu_t^N\), define \(\Delta_E(t)=\mu_t^N(E_N)-\pi_N(E_N)\). For bounded \(E_N\), \(|\Delta_E(t)|\le2\|E_N\|_\infty\|\mu_t^N-\pi_N\|_{\rm TV}\). If \(E_N\) is \(L\)-Lipschitz and the laws have finite first moments, \(|\Delta_E(t)|\le L W_1(\mu_t^N,\pi_N)\). Thus the convergence estimates in the probability chapters give actual relaxation estimates for the chosen observable. Its identification with a gravitational energy density would require a separate calibration.
For a bounded energy observable, total variation gives the error bound directly. For a Lipschitz observable, transporting probability mass gives the corresponding Wasserstein bound. These are the routes by which the convergence chapters turn statistical equilibrium into a quantitative prediction for a measurement.
An Einstein constant enters through an additional field equation. The following flat-space example is a quick way to check that distinction: a constant vacuum stress and a constant geometric term can cancel even when both are nonzero.
Proposition 306 (Stationarity does not determine the Einstein constant)
Conservation of a stress tensor and stationarity of a probability law do not determine a cosmological constant. Even within an Einstein constitutive equation, a static example can have a nonzero constant.
Proof
For the flat metric \(g\), choose a constant \(e\) and \(T_{ab}=-e g_{ab}\). Metric compatibility gives \(\nabla_aT^{ab}=0\). Since \(G_{ab}=0\), the equation \(G_{ab}+\Lambda g_{ab}=\kappa_G T_{ab}\) holds for \(\Lambda=-\kappa_Ge\), which can be nonzero. A constant shift of a statistical energy observable changes its reference but no transition probabilities. Neither conservation nor QSD stationarity selects \(e\) or \(\Lambda\).
Recorded distribution stationarity
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Short-range kernels and constant-curvature metrics#
A small interaction radius tells you which neighborhood a kernel samples. To calculate curvature, you also need to know the limiting metric. Keep those two inputs separate as you examine the explicit anti-de Sitter model below: its negative Einstein constant determines its radius, and substitution verifies the field equation exactly.
Definition 794 (Short-range geometric regime)
The parameter \(\varepsilon_c/L\to0\) denotes a kernel length small relative to a specified geometric length \(L\). Local kernel expansions require the regularity and normalization conditions in Algorithmic Thermodynamics, Graph Cuts, and Boundary Geometry. This scale separation alone imposes no sign on a Ricci tensor.
Theorem 459 (Negative Einstein constant and the AdS model)
For a declared Einstein constant \(\Lambda<0\) in dimension \(d+1\), \(d>1\), the metric constructed in Theorem 456 has radius \(L_{\rm AdS}^2=-d(d-1)/(2\Lambda)\) and satisfies \(G_{ab}+\Lambda g_{ab}=0\). Identifying this metric with a continuum limit of the swarm requires the metric convergence and constitutive equation.
Proof
The cited construction computes \(R_{ab}=-dL_{\rm AdS}^{-2}g_{ab}\) and \(R=-d(d+1)L_{\rm AdS}^{-2}\). Substitution gives \(G_{ab}=d(d-1)g_{ab}/(2L_{\rm AdS}^2)=-\Lambda g_{ab}\). The sign condition fixes the radius of this explicit model, independently of a boundary-pressure interpretation.
The model calculation answers a concrete geometric question: which radius gives the declared Einstein constant? A swarm realization then asks whether its reconstructed metrics and stress tensors converge to the quantities in that calculation. The same care applies to the word “expansion.” A growing cloud variance and a growing infinitesimal spacetime volume have separate definitions.
Definition 795 (QSD regime)
The QSD regime means that the conditioned configuration law is \(\pi_N\), or is close to it in a stated probability metric. The survival probability of the unnormalized killed process may continue to decrease. Stationary one-time statistics permit nonzero stationary probability currents.
Definition 796 (Exploration observable)
An exploration observable is a specified statistic such as the empirical spatial variance or mean pairwise distance. Increasing expectation of this statistic describes spreading in that statistic. A spacetime expansion scalar instead requires a metric and a congruence and is defined by \(\theta=\nabla_a u^a\). Relating the two requires the volume and flux identities of Curvature of the Algorithmic Fitness Metric.
Measured fitness curvature scale
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Quantitative expansion estimates#
Imagine carrying a small bundle of neighboring worldlines. Its volume changes at the rate \(\theta\). Raychaudhuri’s equation says how that rate evolves: curvature, shear, and vorticity all contribute. The comparison theorem keeps every contribution in one forcing term and asks for a lower bound on that whole term.
The function on the right of the theorem starts at zero and approaches \(\sqrt{ad}\). It gives a quantitative growth estimate for every congruence satisfying the forcing bound, throughout the interval on which the congruence remains smooth.
Theorem 460 (A sufficient Raychaudhuri expansion bound)
For a smooth geodesic timelike congruence in \(d+1\) dimensions, define \(F=\omega_{ab}\omega^{ab}-\sigma_{ab}\sigma^{ab}-R_{ab}u^au^b\). If \(F(t)\ge a>0\) and \(\theta(0)\ge0\), then, while the congruence is smooth,
This criterion retains both shear and vorticity. A spreading statistic alone does not supply the curvature and shear bound.
Proof
Raychaudhuri gives \(\dot\theta=-\theta^2/d+F\). Let \(y=\sqrt{ad}\tanh(\sqrt{a/d}\,t)\), which solves \(\dot y=a-y^2/d\), \(y(0)=0\). Then \(w=\theta-y\) satisfies \(\dot w+(\theta+y)w/d=F-a\ge0\). Multiplication by the positive integrating factor and \(w(0)\ge0\) give \(w(t)\ge0\).
The integrating-factor proof is the essential step. Once the comparison solution is subtracted, the difference satisfies a linear differential inequality with a nonnegative source. This fixes the sign of the difference without assuming that shear vanishes.
The de Sitter metric provides a second, explicit calculation. Here we can compute the connection, curvature, and expansion from the same metric and check that they satisfy both Einstein’s and Raychaudhuri’s equations.
Theorem 461 (An explicit de Sitter expansion model)
For \(H>0\), the metric \(ds^2=-dt^2+e^{2Ht}\sum_{i=1}^d(dx^i)^2\) has
and satisfies \(G_{ab}+\Lambda g_{ab}=0\).
Proof
The nonzero Christoffel symbols are \(\Gamma^0_{ij}=Hg_{ij}\) and \(\Gamma^i_{0j}=H\delta^i_j\). Their Ricci contraction gives \(R_{00}=-dH^2\) and \(R_{ij}=dH^2g_{ij}\), hence \(R=d(d+1)H^2\) and the claimed Einstein tensor. For \(u=\partial_t\), \(\theta=\Gamma^i_{i0}=dH\), with zero shear and vorticity. Raychaudhuri reads \(0=-dH^2+dH^2\). This construction proves existence of the geometric model; assigning it to a particular swarm evolution requires a metric identification.
Scale evolution of the executed cloud
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Predictive and dynamical coarse-graining#
A macroscopic description also needs a predictive test. Suppose you replace a complete swarm configuration by a handful of variables. Two microscopic histories can give the same coarse history while carrying different information about what happens next. Closure asks whether that discarded information still improves predictions of the coarse future.
A causal state groups histories by their conditional future law. You can think of it as the complete forecast associated with a history. Information closure asks whether observing the microscopic past improves the forecast of the macroscopic future. Computational closure asks whether the microscopic causal state determines the macroscopic causal state. The distinction matters in the finite-bit example below.
Definition 797 (Predictive causal states)
Two micro-pasts are equivalent when their conditional micro-future laws coincide. The equivalence classes form the predictive causal states \(\Sigma_X\). Define \(\Sigma_Y\) similarly for a deterministic coarse-graining \(Y_t=f(X_t)\). Conditional laws are understood almost surely; for finite alphabets all entropy identities below use finite windows or finite mutual information. The causal state is sufficient for the future by its definition: it records precisely the conditional future law.
Definition 798 (Information closure)
With \(X^-\) denoting the micro-past, \(Y^-\) its coarse-graining, and \(Y^+\) the macro-future, information closure means \(I(Y^+;X^-\mid Y^-)=0\). Equivalently, \(\mathcal L(Y^+\mid X^-)=\mathcal L(Y^+\mid Y^-)\) almost surely.
Definition 799 (Computational closure)
Computational closure means that a well-defined map \(\pi:\Sigma_X\to\Sigma_Y\) sends the causal state of a micro-past to the causal state of its macro-past.
Definition 800 (Preservation of causal equivalence)
Causal closure means that equal micro causal states give equal macro causal states after coarse-graining the past. This is the condition that the map in Definition 799 be well-defined.
Theorem 462 (Closure implications and their converse)
With these definitions, computational and causal closure are equivalent. Information closure implies both. The converse need not hold, including for finite-valued stationary processes.
Proof
A map on equivalence classes is well-defined exactly when representatives from the same class have the same image; this proves the first equivalence. Under information closure the macro-future law given a macro-past equals its law given the micro-past. Equal micro causal states give equal micro-future laws and hence equal macro-future laws. They therefore give equal macro causal states.
For the converse, let \((B_t)_{t\in\mathbb Z}\) be independent fair bits and \(X_t=(B_t,B_{t+1})\), \(Y_t=B_t\). The macro-process is independent, so its causal-state space has one point and computational closure holds. But the micro-past through time \(t\) contains \(B_{t+1}\), while the macro-past does not. Thus \(I(Y_{t+1};X_{\le t}\mid Y_{\le t})=\log2>0\).
In the example, the coarse sequence is a string of independent fair bits, so all coarse pasts give the same forecast. Its causal-state space therefore has one point. Yet a microscopic observation includes the next bit, which improves the prediction of the next coarse observation by one bit. A map between causal states exists even though the discarded information helps prediction.
For a Markov model, there is a direct operational check. Group microscopic states into blocks and add the transition rates into each target block. If those sums agree for all states in a starting block, a macro-observer can use one transition rule for that block. An approximate equality gives the accumulated residual bound in the next proposition.
Proposition 307 (Exact and approximate Markov closure)
For a finite Markov chain with rates \(q(x,z)\) and projection \(f\), the macro-process is Markov for every initial law if \(\sum_{z:f(z)=b}q(x,z)\) depends on \(x\) only through \(f(x)\) for every block \(b\). More generally, suppose a generator \(L_N\) and proposed macro-generator \(\overline L\) satisfy, for a macro test function \(\varphi\),
Then the expected Dynkin residual over an interval of length \(t\) is at most \(t\epsilon_N\) in absolute value. The same bound holds after multiplication by a past-measurable variable of absolute value at most one.
Proof
Under the block-rate condition, \(L(\varphi\circ f)(x)=\sum_b\overline q(f(x),b) [\varphi(b)-\varphi(f(x))]\). The backward equation preserves block-constant functions, so its semigroup induces the claimed macro-transition matrix. For the approximate statement, subtract the two generator integrals in Dynkin’s martingale formula. The remaining integral has absolute value at most \(t\epsilon_N\). The martingale increment has zero expectation, also against a bounded variable measurable at the starting time.
Information in executed region-to-region interactions
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Held-out empirical field closure
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Relating model parameters to observables#
To compare a model with a cosmological measurement, give time and length their physical units and state the field equation being fitted. The first two propositions then become algebra: one converts a dimensionless density parameter into a curvature constant, and the other moves that constant between the geometric and stress sides of Einstein’s equation.
The energy density \(e_{\rm vac}\) and mass density \(\rho_{\rm vac}\) differ by \(c^2\). Keeping that factor visible prevents a statistical energy scale from being inserted into a curvature formula with incompatible units.
Proposition 308 (Cosmological-constant inference within a flat model)
Within a flat Friedmann model with constant vacuum term, define \(\Omega_\Lambda=\Lambda c^2/(3H_0^2)\). Then \(\Lambda=3H_0^2\Omega_\Lambda/c^2\). This inference uses physical time and length units and the specified cosmological model. An algorithmic observable cannot supply \(\Lambda\) without a map to those units and to the model observables.
Proof
Rearrange the definition of \(\Omega_\Lambda\). The units are \([H_0^2/c^2]=\mathrm{length}^{-2}\), as required for curvature.
Proposition 309 (Vacuum energy and pressure in the Einstein convention)
A cosmological term moved to the stress side has \(T^{\rm vac}_{ab}=-\Lambda c^4g_{ab}/(8\pi G_N)\). Its energy density, mass density, and pressure are respectively
Thus \(w=P_{\rm vac}/e_{\rm vac}=-1\) when \(e_{\rm vac}\ne0\).
Proof
Move \(\Lambda g_{ab}\) to the right of the Einstein equation and divide by \(8\pi G_N/c^4\). Compare the resulting tensor with \(T_{ab}=(e+P)u_au_b+Pg_{ab}\). The coefficient of \(u_au_b\) is zero, so \(P=-e\); the coefficient of \(g_{ab}\) determines \(e\).
The relation \(P=-e\) follows from the form of a vacuum stress tensor in the stated convention. It does not require a particular microscopic mechanism. Conversely, observing expansion does not determine that tensor. Milne coordinates give a particularly sharp example: an expanding family of observers lives in flat spacetime with zero cosmological constant.
Theorem 463 (Expansion does not determine the sign of the Einstein constant)
Positive expansion of a geodesic congruence is compatible with both the positive constant in Theorem 461 and a zero constant.
Proof
The de Sitter example was computed above. In Minkowski space introduce Milne coordinates \(T=\tau\cosh\chi\), \(R=\tau\sinh\chi\). The metric is \(-d\tau^2+\tau^2h_{\mathbb H^d}\) and is flat because this is a coordinate change of the Minkowski metric. The comoving congruence has \(\theta=d/\tau>0\), while \(R_{ab}=0\) and \(\Lambda=0\) in vacuum. Consequently expansion alone determines neither a nonzero Einstein constant nor a distance from a statistical QSD. These examples are classical geometric constructions, with no implication for a quantum-gravity existence conjecture.
These examples suggest a practical division of measurements. Estimate a statistical relaxation error from the conditioned law; estimate geometric expansion from a metric and a congruence; test predictive closure from the chosen coarse variables. A physical identification must make their units and dynamical equations agree. The remarks below summarize the assumptions attached to each use.
Remark 302 (Operational equilibrium tests)
A stationary-observable test uses the finite-particle QSD convergence bounds, while a metric-expansion test uses a specified congruence and the volume derivative. Their measurement procedures and hypotheses are separate.
Remark 303 (Statistical spreading and geometric expansion)
The quantitative Raychaudhuri criterion retains the full forcing \(\omega^2-\sigma^2-R(u,u)\). Neither a reward gradient nor an empirical variance increment fixes that forcing without an established metric map.
Remark 304 (Physical calibration)
The energy-reference identity does not resolve a physical vacuum-energy problem. A physical identification requires an effective stress tensor, a constitutive equation, and independently specified units.
Remark 305 (Selection of a stationary state)
Under the proved contraction conditions, the mean-field stationary state is unique; under the finite-particle minorization conditions, the QSD is unique. Other parameter regimes are not assigned multiple stationary states without an existence and nonuniqueness argument.
Dimensionless scales measured from the gas
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Analytical dependencies#
The probability results already proved in this volume supply the statistical part of the argument. Use the theorem for the actual evolution and parameter regime: a finite-particle QSD, a mean-field stationary law, and a spacetime metric answer different questions.
Question |
Analytical input |
Conclusion used here |
|---|---|---|
How quickly does an observable settle? |
Finite-particle QSD theory and entropy convergence, with their stated hypotheses |
Bounds on the conditioned observable excess. |
Is the mean-field stationary law selected uniquely? |
Existence, uniqueness, and attraction in the proved contraction regime. |
|
How is volume growth related to geometry? |
A specified metric, congruence, and flux define the expansion measurement. |
|
Which boundary quantities enter a bulk model? |
The boundary derivative, normalization, and constitutive assumptions remain explicit. |
|
Does a coarse model predict its own future? |
Exact closure implications and quantitative generator residuals. |