Invariant Measures for Soliton Systems Generated by Mealy Automata

Technical Summary: Invariant Measures for Soliton Systems Generated by Mealy Automata

Problem Statement
The paper addresses the characterization of invariant measures for soliton systems described by Mealy automata, specifically focusing on three recently introduced models: BBS-C(2), BBS-S(2), and BBS-V(2). While the Box-Ball System (BBS) has a well-established theory of invariant measures based on soliton decomposition, the structure of invariant measures for these specific Mealy-automaton variants remains largely unexplored. The authors aim to formulate the time evolution of these systems on bi-infinite configuration spaces and derive sufficient conditions under which Bernoulli product measures or two-sided space-homogeneous Markov distributions remain invariant under the automaton dynamics.

Methodology
The authors employ a rigorous mathematical framework combining automata theory, ergodic theory, and statistical physics:

1. Formalism of Mealy Automata: The time evolution is defined by a Mealy automaton $A = (Q, S, \phi, \psi)$ acting on bi-infinite binary sequences $\eta \in \{0, 1\}^\mathbb{Z}$. The state space $Q$ represents a "carrier" that moves across the lattice, picking up and dropping off balls (1s) based on transition ($\phi$) and output ($\psi$) functions.
2. Configuration Spaces: The authors define a specific configuration space $\Omega$ consisting of sequences where arbitrarily long blocks of empty sites (0s) appear infinitely often in both directions. They prove that for the models of interest, this space is invariant under the time evolution map $T_A$.
3. Invariance Criteria:
   • Markov Distributions: A general criterion (Theorem II.12) is derived for a two-sided space-homogeneous Markov distribution $\nu_M$ (characterized by transition probabilities $p$ and $r$) to be invariant. This involves checking a specific ratio of probabilities involving the inverse automaton maps.
   • Bernoulli Distributions: A sufficient condition (Theorem II.13) is established for Bernoulli product measures $\nu_p$ to be invariant. This condition relies on the automaton being "particle-preserving" (conserving a weighted sum of states and inputs) and satisfying specific structural assumptions (bijectivity, existence of a unique fixed point for the empty input).
4. Soliton Analysis: The authors analyze the soliton dynamics of the three models, defining soliton decomposition algorithms, calculating bare velocities, and determining phase shifts resulting from soliton interactions.

Key Contributions and Results

• General Framework: The paper establishes a general theory for the invariance of Markov and Bernoulli measures for automata satisfying conditions (A1)–(A3) (bijectivity, specific fixed-point behavior, and reset properties).
• Application to Three Models:
  • BBS-S(2) and BBS-V(2): The authors prove that for these two models, Bernoulli product measures $\nu_p$ are invariant for any $p \in (0, 1)$. This is achieved by verifying that these automata are particle-preserving and satisfy the structural conditions required by Theorem II.13.
  • BBS-C(2): For this model, Bernoulli measures are not generally invariant. Instead, the authors prove that the broader class of two-sided space-homogeneous Markov distributions (where the transition probability $r$ is not necessarily $1-p$) is invariant. This result is derived by explicitly verifying the invariance condition (Theorem II.12) through a detailed case analysis of the inverse dynamics.
• Soliton Characteristics:
  • Velocities: The bare velocities are computed: $v=2$ for size-$k$ solitons in BBS-C(2); $v=1$ and $v=2$ for "11" and "10" solitons in BBS-S(2); and $v_m = (m+1)/m$ for $m$-solitons in BBS-V(2).
  • Phase Shifts: The phase shifts resulting from soliton interactions are calculated. Notably, the phase shifts in BBS-S(2) and BBS-V(2) are found to be non-antisymmetric ($\Phi(J, K) \neq -\Phi(K, J)$), distinguishing them from the standard BBS.
• Marginal Invariance: The paper demonstrates that for all three models, even when the full measure is not invariant, the marginal distribution at each site (the density of particles) remains unchanged under time evolution, a direct consequence of the particle-preserving property.

Significance
The paper provides a foundational framework for studying invariant measures, generalized Gibbs ensembles (GGE), and generalized hydrodynamic (GHD) behavior in Mealy-automaton soliton systems. By identifying specific invariant measures (Bernoulli for BBS-S(2)/BBS-V(2) and Markov for BBS-C(2)), the authors enable the application of GGE and GHD theories to these systems. The results suggest that these three models serve as convenient frameworks for testing ideas related to integrable systems described by automata, particularly regarding how different types of invariant measures (product vs. Markov) arise from specific automaton structures. The non-antisymmetric phase shifts observed in BBS-S(2) and BBS-V(2) highlight unique hydrodynamic behaviors distinct from the classical BBS.