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What Is a Macrostate? Subjective Observations and Objective Dynamics

Cosma Rohilla Shalizi, Cristopher Moore

arXiv:cond-mat/0303625v1cond-mat.stat-mech

TL;DR

The paper asks how to reconcile macrostates as observer-dependent choices with their apparently objective physical dynamics. It connects macrostates to causal states, refining observational partitions to obtain Markovian predictors; these states provide objective descriptions while depending on the questions observers choose to ask.

Abstract

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We consider the question of whether thermodynamic macrostates are objective consequences of dynamics, or subjective reflections of our ignorance of a physical system. We argue that they are both; more specifically, that the set of macrostates forms the unique maximal partition of phase space which 1) is consistent with our observations (a subjective fact about our ability to observe the system) and 2) obeys a Markov process (an objective fact about the system's dynamics). We review the ideas of computational mechanics, an information-theoretic method for finding optimal causal models of stochastic processes, and argue that macrostates coincide with the ``causal states'' of computational mechanics. Defining a set of macrostates thus consists of an inductive process where we start with a given set of observables, and then refine our partition of phase space until we reach a set of states which predict their own future, i.e. which are Markovian. Macrostates arrived at in this way are provably optimal statistical predictors of the future values of our observables.

I. WHAT’S STRANGE ABOUT MACROSTATES, OR, IS IT JUST ME?

Macrostates are difficult to treat as purely subjective or purely objective: different observables induce different partitions, while multiple levels of description can be empirically valid. The paper proposes refining observational partitions until their dynamics are Markovian and connects these optimal predictors to computational mechanics’ causal states.

I. WHAT’S STRANGE ABOUT MACROSTATES, OR, IS IT JUST ME?

  • Macrostates are hard to classify as purely subjective or objective when different observable sets induce different partitions and multiple validated description levels coexist.Thermodynamic and hydrodynamic theories are both validated for many systems, despite representing different levels of description.Source: §I · p. 2
  • Good macrostates are the coarsest refinements of chosen observational states whose dynamics are Markovian.The refinement process adds observable variables when the initial partition fails to predict its own future.Source: §I · p. 2

II. COMPUTATIONAL MECHANICS AND CAUSAL STATES

  • Computational mechanics finds a minimal model of a stochastic process by grouping histories that imply the same conditional distribution over futures.The resulting causal states capture predictive patterns in observed time series.Source: §II · p. 3
  • Causal states are prescient and, except for a set of measure zero, form the coarsest prescient partition of histories.A prescient partition retains the full mutual information between past and future.Source: §II · p. 3
  • Among prescient partitions, causal states uniquely minimize statistical complexity, defined as the entropy of the partition.Here complexity measures information encoded about the past, so minimizing it selects the least-complex predictor.Source: §II · p. 3
  • The causal-state sequence is Markovian while retaining all information from the original process that is relevant to its future.The process’s statistical complexity can exceed its predictive information; for first-order Markov processes, Cµ = E + hµ.Source: §II · p. 4

III. CAUSAL STATES FROM COARSE-GRAINED OBSERVATIONS

The paper constructs causal states from histories of observations of a system whose microstate evolves as a Markov chain. These states induce a phase-space partition with Markovian dynamics, while their relation to the original observational partition can vary.

III. CAUSAL STATES FROM COARSE-GRAINED OBSERVATIONS

  • The underlying phase-space evolution is represented as a possibly stochastic operator on microstates, with the microstate sequence forming a Markov chain.The setup excludes hidden variables and does not require an invariant ensemble distribution.Source: §III · p. 4
  • An observation function partitions phase space, but the resulting observed process need not itself be Markovian.The observation function maps each microstate to a value in the observation space.Source: §III · p. 4
  • Causal states of observational histories induce a phase-space partition whose process is Markovian and sufficient for optimal prediction of future observations.The causal-state partition coarsens the partition induced by infinite-length observational histories.Source: §III · pp. 4–5
  • The causal and observational partitions may coincide, or the causal partition may be finer, coarser, or incomparable.The section identifies these as four possible relationships between the partitions.Source: §III · p. 5

A. The Observables Define a Macrostate

When current macroscopic values determine causal state, the observables themselves define a macrostate. Such macrostates have Markovian, statistically reproducible dynamics, and their present values suffice for optimal prediction of future macrovariables.

A. The Observables Define a Macrostate

  • When causal equivalence is determined by the current observable values, the macrostate and causal-state partitions coincide.The example given is pressure, volume, and temperature for an ideal gas near equilibrium.Source: A. The Observables Define a Macrostate · p. 5
  • These macrostates are Markovian and statistically reproducible, and no prediction of future macrovariables improves on using their present values.This holds in the case where the observables define the causal state.Source: A. The Observables Define a Macrostate · p. 5
  • Statistical complexity is the amount of information about the microstate contained in the macrovariables.For first-order Markov macrostates, predictive information equals statistical complexity minus entropy rate.Source: A. The Observables Define a Macrostate · p. 5

B. The Causal States Are Finer than the Macrostates

Causal states can be reached by refining observed macrovariables until their dynamics are Markovian, and factor variables can provide that refinement. If those factors are inaccessible, causal states can instead be constructed from observational histories.

B. The Causal States Are Finer than the Macrostates

  • Knowing the causal state gives Markovian time-evolution and optimal predictions, whereas the original macrovariables may require information about their history.A refinement of the original partition can make the macrovariables causally complete.Source: B. The Causal States Are Finer than… · p. 5
  • A minimal factor partition Z can refine the original observables F so that observing (F, Z) is equivalent to observing the causal state.This yields a causally complete set of macrovariables.Source: B. The Causal States Are Finer than… · pp. 5–6
  • Different factor partitions have the same uncertainty remaining after the original observables are known, and their information is relevant to the causal state.The text states that H[Z|S] is independent of the factor partition chosen.Source: B. The Causal States Are Finer than… · p. 6
  • Factor partitions may be experimentally inaccessible, but causal states built from observational histories can still eliminate memory effects.This provides an alternative when no practical measurement procedure can access a factor partition.Source: B. The Causal States Are Finer than… · p. 6
  • In fluid-jet dispersion, refining the initial motion-character partition was reported to eliminate memory effects that complicated analysis.The cited work includes Abel et al. (2002) and Lacorata et al. (2001).Source: B. The Causal States Are Finer than… · p. 6

1. An Apparent Counterexample: Disordered Materials

Disordered materials pose a challenge because their dynamics span many timescales and may not reach equilibrium on laboratory timescales. For spin glasses, the paper proposes tracking occupancy probabilities across accessible metastable minima to handle aging without requiring the history of probability seepage.

1. An Apparent Counterexample: Disordered Materials

  • Disordered materials display distinct dynamics across immense timescale ranges, and equilibrating low-temperature spin glasses may require geologically long waits.The passage says the longevity needed for aged equilibrium samples is impractical for ordinary laboratory work.Source: §1 · p. 6
  • Frustration creates many energy minima and barriers of varying heights, while crossing a barrier takes a time generally exponential in its height.On a given timescale, sufficiently high barriers are effectively uncrossable, confining the spin glass to a region of phase space.Source: §1 · pp. 6–7
  • Within the accessible phase-space region, local minima act as metastable states whose lifetimes and relative probabilities reflect surrounding barrier heights.Source: §1 · p. 7
  • Spin-glass macrovariables can be occupancy probabilities over accessible metastable minima, allowing forward extrapolation from the current distribution without knowing the history of its seepage.The passage notes that experiments study aging samples and that aging properties are of interest.Source: §1 · p. 7

C. The Macrostates Are Finer than the Causal States

Some distinctions between macrostates do not improve predictions: variables may be irrelevant, encode an unphysical distinction, or be unpredictable at the observation scale. Coarsening such distinctions can preserve predictive power, and at extreme coarse scales the process may become IID.

C. The Macrostates Are Finer than the Causal States

  • Variables whose values are statistically independent of future events given other current variables add no predictive information and can be removed.The passage notes that deterministic functions of other variables induce the same phase-space partition when added.Source: C. The Macrostates Are Finer than the… · p. 7
  • A nematic director's opposite orientations are physically identified, so treating them as one observation coarsens the partition without losing predictive power.The passage explains that otherwise the observed sign is a matter of chance.Source: C. The Macrostates Are Finer than the… · p. 7
  • At the available timescale and resolution, a randomized variable can wash out of the predictive partition; in the extreme, observations become IID and the partition is trivial.For the example, pressure and internal energy measured yearly in an ideal gas have no predictive value.Source: C. The Macrostates Are Finer than the… · p. 7
  • A variable that is effectively IID over long times or at coarse resolution can still contain substantial predictive information at finer scales.Source: C. The Macrostates Are Finer than the… · p. 7

D. The Physical Meaning of the Causal States

Causal-state macrostates make dynamics objective while retaining the dependence of descriptions on what is observed. They can clarify emergence across description levels and, in the argon example, predict macrovariables more efficiently than microstates.

D. The Physical Meaning of the Causal States

  • Starting from observational variables, one can construct causal states and derive minimal coarse-grained observations that optimally predict the original observables.The resulting variables can be added to, retained from, or used to replace the original observations.Source: D. The Physical Meaning of the Causal… · p. 8
  • Causal-state macrostates are Markovian and have a Gibbs distribution over sequences without requiring an equilibrium assumption.Their statistical complexity measures how much information the macrovariables contain about the microstate.Source: D. The Physical Meaning of the Causal… · p. 8
  • The initial choice of observables is subjective, but the subsequent causal-state construction is fixed by the objective microdynamics.The causal states may require adding variables, leaving them unchanged, or coarsening them.Source: D. The Physical Meaning of the Causal… · p. 8

IV. LEVELS OF DESCRIPTION AND EMERGENCE

  • Different measurement partitions can yield coherent causal accounts at different description levels, although coarsening a causal partition does not generally preserve Markovianity.The causal states of a coarsened measurement are well-defined and cannot be finer than the states of the finer partition.Source: §IV · p. 8

A. Equilibrium Fluctuations and a Definition of Emergence

  • 0.87 versus 0.5: macrostates have higher predictive efficiency than microstates in the argon example, despite being predicted over a longer timescale.At the same time resolution, macrostate efficiency would be essentially one; at the macro timescale, microstate efficiency would be essentially zero.Source: A. Equilibrium Fluctuations and a… · p. 9
  • Emergence is a coarse-graining whose variables are more predictively efficient, a criterion used to characterize thermodynamics as emerging from statistical mechanics.Some coarse-grainings can instead be less efficiently predictable, a condition termed “submergence.”Source: A. Equilibrium Fluctuations and a… · p. 9

B. Hydrodynamics and Levels of Description

Hydrodynamic and thermodynamic descriptions can coexist because they answer different predictive questions, and their relationship is generally one of emergence. Cellular-automaton modeling illustrates how coarse-graining seeks simple representations that retain information relevant to large-scale dynamics.

B. Hydrodynamics and Levels of Description

  • Hydrodynamic descriptions retain local densities needed to predict local observations, while thermodynamic descriptions suffice for global averages or sums.The two levels coexist because they address different questions, not because one is more objective.Source: B. Hydrodynamics and Levels of… · p. 9
  • The relationship between hydrodynamic and thermodynamic descriptions is generally one of emergence, and local causal states generally form a Markov random field.Whether local causal states always form a Markov random field remains an open question; such fields have Gibbs distributions over configurations.Source: B. Hydrodynamics and Levels of… · p. 9, C. Building Coarse Grainings in… · p. 9

C. Building Coarse Grainings in Cellular Automata and Lattice Gases

  • Cellular automata have discrete time and space, with each cell updated from its state and a fixed neighborhood; they model natural phenomena and dynamic spin systems.Their local interactions make them examples for constructing coarse-grainings in practice.Source: C. Building Coarse Grainings in… · pp. 9–10
  • Coarse-graining cellular automata discards microscopic detail to find simple but accurate representations that retain information relevant to selected large-scale dynamics.Spatial computational mechanics provides tools for identifying and filtering low-information patterns in favor of information-rich structures.Source: C. Building Coarse Grainings in… · p. 10

V. WHAT IS NOT BEING SAID

The paper leaves several questions about why particular macroscopic variables are common or reusable unresolved. Its conclusion makes macrostates depend on observable choices while holding their causal states fixed by the system’s dynamics.

V. WHAT IS NOT BEING SAID

  • Why extensive quantities are often good macrovariables remains unresolved; large-deviation behavior under approximate subsystem independence is only a possible explanation.When approximate independence fails, the authors say good macrovariables may be non-extensive and require Tsallis rather than usual central-limit-theorem statistics.Source: §V · p. 10
  • Why good macrovariables are often derivatives of thermodynamic potentials remains unanswered; selection pressure on candidate potentials is offered as a tentative explanation.Source: §V · p. 10
  • The reuse of macrovariables such as temperature across different systems remains unexplained, while order parameters are generally not reusable.Source: §V · p. 10
  • The approach assumes coarse-grained observations may be treated statistically, while the legitimacy of that assumption for large, locally unstable mechanical systems remains unresolved.The authors regard maximum-entropy justification as philosophically contested and point instead to ergodic theory and generic initial conditions as a possible direction.Source: §V · p. 10

VI. CONCLUSION

  • The construction ends with a minimal set of variables that optimally predicts the macrovariables of interest and is causally complete.Source: §VI · p. 11
  • The initial macrovariables reflect observational abilities and predictive interests, but the resulting causal states are fixed by objective microphysical dynamics.Different initial variables generally lead to different causal states, which are sometimes—but not always—related by emergence.Source: §VI · p. 11

APPENDIX A: Information Theory

The appendix defines information-theoretic quantities used to describe uncertainty and dependence. Entropy measures average information, while joint, conditional, and mutual information characterize how variables relate.

APPENDIX A: Information Theory

  • Shannon entropy is the average information needed to specify the value of a discrete random variable.It is the smallest average number of bits needed for that specification.Source: Appendix A · p. 11
  • Joint entropy is defined for two variables, is no greater than the sum of their entropies, and equals that sum exactly when they are statistically independent.Source: Appendix A · p. 11
  • Conditional entropy H[X|Y] is the average information remaining in X when Y is known.For a particular value y, H[X|Y = y] measures the information needed to specify X within the sub-ensemble where Y = y.Source: Appendix A · pp. 11–12
  • Mutual information is the amount by which knowledge of one variable reduces uncertainty about the other.Source: Appendix A · p. 12

APPENDIX B: Markov processes

A first-order Markov process has no dependence of its next state on earlier history once its current state is known. The appendix illustrates this property with several processes and shows how expanding observables can make a process Markovian.

APPENDIX B: Markov processes

  • A first-order Markov process is one in which the next state depends on the entire past only through the current state.Source: Appendix B · p. 12
  • A deterministic system, fair coin flips, and Brownian motion are given as examples of Markov processes.Source: Appendix B · p. 12
  • A particle’s position alone is non-Markovian under constant-velocity motion because its next position depends on the previous position as well as the current one.Source: Appendix B · p. 12
  • Including both position and velocity in the particle’s observables makes the process first-order Markovian.Source: Appendix B · p. 12

APPENDIX C: Partitions

The appendix formalizes partitions, equivalence relations, and refinement operations. It also establishes that a refinement can be represented by combining the original partition with a minimal factor partition.

APPENDIX C: Partitions

  • A partition divides a set into mutually exclusive, jointly exhaustive cells, and each point belongs to an equivalence class under an equivalence relation.Source: Appendix C · p. 12
  • Every partition corresponds to an equivalence relation whose equivalence classes are the partition’s cells, and vice versa.Source: Appendix C · p. 12
  • A function induces a partition by grouping points with the same value, while each equivalence relation can be represented by functions assigning unique labels to its classes.Source: Appendix C · p. 12
  • Refinement makes a partition finer, and the product of two partitions—formed from intersections of their cells—is a refinement of both.The identity partition has singleton cells, while the trivial partition has one cell containing the whole set.Source: Appendix C · p. 12
  • Any refinement B of A can be expressed as the product of A and a minimal factor partition C, with no smaller-cell partition serving as that factor.Minimal factors are incomparable to A.Source: Appendix C · p. 13
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