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Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide

R. A. Blythe, M. R. Evans

arXiv:0706.1678v1cond-mat.stat-mech

TL;DR

The review addresses limited knowledge of nonequilibrium steady-state microstate statistics. It develops matrix product methods for exact macroscopic calculations and surveys their physical consequences, connections, and scope boundaries.

  • Problem

    Nonequilibrium systems lack a generally known description of steady-state microstate statistics despite many analytical approaches.

  • Method

    The review explains matrix product steady-state representations and how they yield exact currents, density profiles, correlations, and fluctuation distributions.

  • Results

    The reviewed calculations expose boundary-induced phase transitions, shock fronts, kinematic waves, and transitions between distinct current-density regimes.

  • Takeaways & Limitations

    Matrix product methods provide an analytically tractable route from mesoscopic nonequilibrium dynamics to macroscopic steady-state properties and equilibrium-statistical-physics connections.

  • Takeaways & Limitations

    For open boundaries, a matrix product solution for driven extended objects has proved elusive because boundary-term cancellations are difficult to arrange.

Abstract

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We consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model (among other things) biological transport and traffic flow. We begin with a broad overview of this class of driven diffusive systems - which includes exclusion processes - focusing on interesting physical properties, such as shocks and phase transitions. We then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form. In addition to a gentle introduction to this matrix product approach, how it works and how it relates to similar constructions that arise in other physical contexts, we present a unified, pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed. We also review a number of more advanced topics, including nonequilibrium free energy functionals, the classification of exclusion processes involving multiple particle species, existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled. We conclude with a brief discussion of open problems for future research.

1. Introduction

The review introduces nonequilibrium steady states, explains why their microscopic statistics remain difficult to determine, and presents matrix product methods for exact steady-state analysis. It connects mesoscopic stochastic models to macroscopic phenomena and equilibrium statistical physics while defining the review’s scope and aims.

  • What is a nonequilibrium steady state?: Nonequilibrium steady states have time-independent observables but sustain irreversible exchanges and a nonzero flux through the system.The flux is constant over space and time after initial transients.
  • What is a nonequilibrium steady state?: Unlike equilibrium systems with Gibbs-Boltzmann statistics and no steady-state currents, nonequilibrium systems generally do not satisfy detailed balance.Equilibrium dynamics are reversible, whereas steady currents are characteristic of nonequilibrium systems.
  • What is a nonequilibrium steady state?: A coherent account of macroscopic nonequilibrium steady-state properties remains lacking, especially for systems driven far beyond the linear-response regime.Existing approaches include macroscopic fluctuation theories, hydrodynamic equations, and fluctuation theorems.
  • Purpose of this review: The review focuses on analytically tractable mesoscopic models whose steady-state microstate distributions have matrix product form.The approach is used to calculate currents, density profiles, correlations, and macroscopic fluctuation distributions exactly.
  • Purpose of this review: Matrix product calculations expose boundary-induced phase transitions and shock fronts while revealing conceptual connections with equilibrium statistical physics.Noncommuting matrices can also encode correlations between occupancies at different sites.
  • Purpose of this review: The review aims to explain exact-model insights, provide a self-contained account of analytical tools, and collect recent progress on static steady-state properties.It excludes some dynamical topics to keep the review focused on static properties.

2. An overview of driven diffusive systems

Driven diffusive systems exhibit shocks and boundary-controlled phase transitions, while the ASEP provides a setting where these behaviors can be analyzed through kinematic waves, mean-field theory, and exact matrix-product methods.

  • Driven diffusive systems: The ASEP is a continuous-time exclusion process in which particles hop asymmetrically on a one-dimensional lattice, with either conserved particle number on a ring or reservoir-driven boundary dynamics.On a ring, all allowed configurations with fixed particle number are equally likely in the steady state; open boundaries inject particles at rate α and remove them at rate β.
  • Driven diffusive systems: The ASEP on a ring has relaxation time T ∼ N^z with dynamic exponent z = 3/2, while the symmetric exclusion process has z = 2.
  • Kinematic waves and shocks: Kinematic waves describe density patches propagating with group velocity v_g(ρ); when v_g decreases with density, faster low-density regions overtake slower high-density regions and form shocks.After shock formation, first-order density equations require second-order spatial derivatives to describe the shock profile.
  • Kinematic waves and shocks: For α, β < 1/2, boundary-launched waves meet in a shock: β > α yields a bulk density α, α < β yields a bulk density 1 − β, and α = β produces a diffusing stationary shock.When α = β < 1/2, the shock is equally likely to occur anywhere in the system.
  • Phase behavior: When α, β > 1/2, boundary waves do not penetrate, and the system adopts maximal current with density ρ_m = 1/2; the current is J = 1/4 + O(N^-2).The maximal-current density corresponds to zero kinematic-wave velocity and requires a diffusive contribution to the current.
  • Phase behavior: The extremal-current principle selects J0(ρ̄) as a maximum when ρ_R < ρ_L and a minimum when ρ_L < ρ_R, organizing the ASEP phase diagram.Generating-function singularities reproduce the same phase regions exactly, while matrix-product constructions provide the steady-state framework for exact calculations.

3. Detailed analysis of the ASEP with open boundaries

The section proves the open-boundary ASEP matrix-product weights are stationary by organizing master-equation terms into cancellations. It then generalizes this mechanism algebraically and derives exact expressions for observables.

  • Matrix-product construction: The matrix-product expressions for configuration weights, densities, and currents are evaluated using reduction relations after stationarity is established.The proof begins by showing that the proposed weights solve the master equation.
  • Domain-based proof: On a ring, each particle domain contributes one gain and one loss process, so uniform weights satisfy the master equation through pairwise balance.This cancellation differs from detailed balance because the incoming and outgoing configurations need not be identical.
  • Domain-based proof: With open boundaries, unequal entry and exit processes require partial cancellation among domain, boundary, and reservoir terms.Boundary-attached domains leave residual terms that cancel with neighboring domains or the opposite boundary.
  • Exact observables: Using the reduction relations and boundary normalization, the construction yields exact expressions for steady-state quantities such as density profiles.For 1 ≤ i < N, the density is expressed through normalization ratios, with a special formula at i = N.
  • Algebraic proof: Auxiliary matrices convert local master-equation contributions into telescoping differences, whose sum vanishes when bulk and boundary relations agree.The sufficient conditions are encoded by equations (3.21)–(3.23).
  • Algebraic proof: Scalar auxiliary choices ˜D = −1, ˜E = 1, and ˜A = 0 reduce the general conditions to the three ASEP reduction relations.The same algebraic proof can be generalized to models with more general bulk dynamics.

3.4. Asymptotic analysis of the current and density profiles

The section obtains the ASEP phase diagram and density profiles from large-N generating-function asymptotics. Dominant poles and algebraic singularities reproduce the mean-field phases while revealing exact boundary-decay behavior and phase-transition structure.

  • Generating-function method: Generating functions provide a systematic route to large-N asymptotics of the current, normalization, and density profiles.The method extracts coefficient asymptotics from singularities of the generating functions.
  • Generating-function method: The dominant singularity nearest the origin determines the asymptotic normalization, with a square-root singularity and boundary-dependent poles controlling different phases.The poles occur on the relevant branch only under corresponding conditions on α or β.
  • Current asymptotics: J = α(1 −α) in the low-density phase and the particle-hole-symmetric expression holds in the high-density phase.In the maximal-current phase, J = 1/4.
  • Current asymptotics: J = 1/4 + O(N−2) in the maximal-current phase, confirming the mean-field asymptotic current in all three phases.The exact generating-function analysis establishes the mean-field current predictions asymptotically.
  • Density profiles: Generating-function singularities determine whether density corrections are exponential, power-law, or linear across the ASEP phases.Mean-field theory gets the bulk density right but can predict an incorrect decay exponent.
  • Exact phase diagram: The exact phase diagram adds two high- and low-density sub-phases distinguished by different boundary-density decay forms.The decay length diverges on approach to the maximal-current phase, while coexistence at the high-/low-density boundary resembles a first-order transition.
  • Equilibrium connection: The ASEP normalization is also the partition function of a one-transit walk with contact fugacities 1/α and 1/β.This mapping identifies low-, high-, and maximal-current phases with adsorption from above, adsorption from below, and desorption.

4. ASEP on a ring with two particle species

The two-species ASEP on a ring extends the matrix-product construction to vacancies, first-class particles, and a second particle species. Fixed particle numbers and grand-canonical analysis produce defect-induced phases, including a particle-coexistence phase with a localized shock.

  • Model and ensemble: A two-species exclusion process contains species 1 particles, species 2 particles, and vacancies, with particle numbers conserved on the ring.The case α = β = 1 identifies species 1 and species 2 with first- and second-class particles.
  • Matrix-product state: Rotational invariance motivates representing ring configuration weights by tr(Xτ1Xτ2 · · · XτN).D, E, and A denote matrices for species 1 particles, vacancies, and species 2 particles.
  • Stationarity proof: Auxiliary matrices generate pairwise cancellations, and scalar choices reduce the nine local conditions to the three matrix reduction relations.This proves stationarity of the two-species matrix-product weights.
  • Grand canonical ensemble: The grand-canonical fugacity u controls the density of occupied sites and can also be generated dynamically by non-number-conserving rules.The canonical and grand-canonical descriptions are used to analyze fixed and fluctuating particle numbers.
  • Phase diagram: Four thermodynamic phases arise for a single defect, including power-law, exponential, and particle-coexistence regimes.The phase diagram and density profiles are organized by the values of α, β, and the bulk density ρ.
  • Phase coexistence: In the particle-coexistence phase, low density β and high density 1 −α meet at a shock whose position is fixed by the conserved bulk density.The first-class current is J = ρ(α −β) + β(1 −α) in this phase.
  • Multiple defects: Two second-class particles exhibit a 1/r3/2 separation distribution, implying a weak bound state with diverging mean separation.The result corresponds to an effective attractive interaction.

5. The partially asymmetric exclusion process with open boundaries

The PASEP extends exclusion dynamics with leftward and rightward hopping, while its matrix-product solution yields phase behavior, currents, and density profiles through asymptotic analysis.

  • 5. The partially asymmetric exclusion process with open boundaries: Particles hop left and right with rates p and q under hard-core exclusion, with open-boundary driving.The time unit is fixed by setting p = 1.
  • 5. The partially asymmetric exclusion process with open boundaries: The model is physically connected to nonequilibrium surface growth, KPZ universality, reverse-bias transport, and ballistic reaction dynamics.The KPZ nonlinearity is proportional to 1 −q, while q > 1 can oppose the boundary-imposed current.
  • 5. The partially asymmetric exclusion process with open boundaries: The stationary distribution uses particle and hole matrices D and E contracted with boundary vectors, with algebraic relations enabling exact calculations.The master equation is decomposed into boundary and bulk terms that telescope through auxiliary matrices.
  • 5. The partially asymmetric exclusion process with open boundaries: The matrix product Z_N = ⟨0|(D + E)^N|0⟩ maps to weighted nonnegative Motzkin paths with two types of horizontal segments.Up-down pairs carry height-dependent weights c_n, while horizontal segments have weights 1 + aq^n and 1 + bq^n.
  • 5. The partially asymmetric exclusion process with open boundaries: The PASEP has three phases, with normalization asymptotics selected by poles and boundary parameters a and b.When both a and b exceed 1, the dominant contribution depends on whether a > b or a < b; corresponding currents follow from matrix expressions.
  • 5. The partially asymmetric exclusion process with open boundaries: For q > 1, the bulk bias opposes the boundary current, producing a single reverse-bias phase whose current vanishes asymptotically.Its density profile is conjectured to contain particle-rich and hole-rich domains separated by an interface.

6. Macroscopic density profiles for open boundary ASEP

The review develops nonequilibrium free energy functionals for open exclusion systems, showing how boundary driving produces nonlocal correlations and altered fluctuation behavior.

  • 6. Macroscopic density profiles for open boundary ASEP: The steady-state distribution permits calculation of macroscopic quantities beyond currents and density profiles, including entropy and density-profile fluctuation functionals.These functionals describe the probability of macroscopic profiles relative to the most likely profile.
  • 6. Macroscopic density profiles for open boundary ASEP: For the open SSEP, the free energy functional is derived from its matrix-product steady-state distribution and boundary-rate dynamics.The model has symmetric unit-rate bulk hopping and boundary entry and exit rates α, β, γ, and δ.
  • 6. Macroscopic density profiles for open boundary ASEP: The equilibrium condition αβ = γδ is required for an equilibrium steady state, because boundary-cycle rate products must balance.When this condition holds, the equilibrium free-energy functional is recovered and the relevant profile is linear.
  • 6. Macroscopic density profiles for open boundary ASEP: Relaxing αβ = γδ makes the free-energy density nonlocal because the matrix-product distribution induces long-range correlations between density boxes.The auxiliary function σ(x) can depend on the entire density profile through a differential equation.
  • 6. Macroscopic density profiles for open boundary ASEP: Nonequilibrium fluctuations are generally more suppressed than equilibrium fluctuations, with F[ρ] ≥ F_eq[ρ] under the stated comparison.For the PASEP, fluctuation behavior can instead be enhanced when boundary densities oppose the bulk bias.
  • 6. Macroscopic density profiles for open boundary ASEP: PASEP free-energy functionals are q-independent after thermodynamic rescaling, but the limits N →∞ and q →1 do not commute.A weakly asymmetric scaling 1 −q = λ/N restores explicit q- or λ-dependence.
  • 6. Macroscopic density profiles for open boundary ASEP: In the maximal-current phase, density fluctuations are non-Gaussian rather than described by Gaussian statistics.This is supported by explicit calculations for the totally asymmetric case q = 0.
  • 6. Macroscopic density profiles for open boundary ASEP: The additivity approach requires an additive function and local-equilibrium property, but the appropriate additivity form is not known a priori for general systems.The SSEP functional has also been derived through a purely macroscopic formalism.

7. Two-species Models with quadratic algebra

This section classifies two-species exclusion dynamics admitting matrix-product steady states with quadratic algebras. It connects algebraic consistency conditions to stationary weights and physical phenomena including phase separation and nonconserving transitions.

  • Algebraic framework: The analysis searches a restricted class of two-species models whose matrix products reduce systematically through associative quadratic relations.The reduction must be order-independent, yielding PBW-type algebras.
  • Algebraic framework: Stationarity follows when exchange-rate relations are represented by matrices and auxiliary quantities satisfying the local cancellation condition.The resulting matrix-product weights solve the stationary master equation.
  • Classification: The classification imposes six hopping-rate conditions obtained by requiring each term in the consistency equation to vanish.Solutions are organized according to how many auxiliary scalars x_i are zero.
  • Physical models: For symmetric two-species exclusion on a ring, all allowed configurations are equally likely despite the matrix-product representation.The matrices can be reduced so that periodic strings have identical weights.
  • Physical models: The cyclic ABC model with q < 1 develops strong phase separation into pure domains ordered ABC, while a weakly asymmetric scaling produces a phase transition.For equal species numbers, the matrix product permits exact steady-state calculations.
  • Extensions: Closed conservative segments satisfy detailed balance, whereas nonconserving extensions can impose prescribed sector weights while retaining nonequilibrium dynamics within sectors.The construction augments sector-changing processes so detailed balance holds between sectors but not necessarily inside them.
  • Extensions: For a model without conserved particle numbers, varying w causes a vacancy-density transition: vacancies vanish for w < 1 and become finite for w > 1.The transition is identified from the asymptotic normalization behavior and the changing tendency to create or eliminate charges.

8. Multispecies Models with quadratic algebra

This section extends quadratic-algebra matrix-product methods from two species to arbitrarily many species and disordered hopping rates. The extensions capture condensation, traffic jams, overtaking, and changes in open-system phase structure.

  • Multispecies extensions: Two families of two-species dynamics generalize to arbitrarily many particle species, preserving matrix-product methods based on quadratic algebras.The section develops models with species-dependent hopping and overtaking dynamics.
  • Species-dependent rates: For particles with species-dependent forward and reverse rates, configuration weights use traces of alternating particle matrices and vacancy powers.The weight is written as Tr[D1 E^n1 D2 E^n2 ... DM E^nM].
  • Species-dependent rates: When only forward hops occur, condensation can place an extensive number of vacancies before the slowest particle.The resulting macroscopic gap forms a traffic jam behind that particle.
  • Overtaking dynamics: In the many-species overtaking model, particles exchange according to differences in their velocities, generalizing the two-species construction.The matrix relations recover the usual ASEP algebra when only one particle species remains.
  • Overtaking dynamics: On a periodic lattice, a scalar representation makes all allowed configurations equally likely in the overtaking model.Open boundaries require additional rate choices and matrix constraints.
  • Disordered velocities: For continuously distributed particle velocities, the open-system phase diagram depends on the distribution near its minimum velocity.If l[σ] < 0 all three phases remain; if l[σ] ≥ 0 the high-density phase is suppressed.

9. More complicated matrix product states

The review extends matrix product steady states beyond scalar auxiliaries and develops existence constructions for open systems with arbitrary nearest-neighbour interactions. It also examines non-conserving and multi-species models, while highlighting limitations of constructive calculations and periodic-boundary proofs.

  • Existence proofs: For open boundaries and arbitrary nearest-neighbour interactions, an existence proof constructs matrix product representations using configuration-basis auxiliary objects.The construction establishes consistency but does not provide convenient reduction relations or explicit matrices without prior knowledge of the steady-state weights.
  • General cancellation mechanism: Matrix product weights f(τ1,...,τN)=⟨W|Xτ1···XτN|V⟩ become stationary when auxiliary matrices enable pairwise cancellation of bulk and boundary terms.The cancellation relations generalize the ASEP algebra and yield a zero right-hand side of the master equation.
  • Representational freedom: The construction shows that one stationary master equation can correspond to many choices of matrices and auxiliaries, and ASEP-style reduction relations are sufficient rather than necessary.Auxiliary matrices generated from the stochastic-process generators need not imply convenient reduction relations.
  • Periodic boundaries: Periodic systems remain harder because trace cyclicity can impose global particle-number constraints, obstructing an existence proof parallel to the open-boundary construction.The ABC model is consistent only when the numbers of the three species are equal.
  • Non-conserving models: Finite-dimensional matrix product states also occur in non-conserving reaction-diffusion models, including diffusion, coagulation, and decoagulation, with a four-dimensional representation found for one example.For the closed segment, excluding the inactive empty-lattice steady state requires N-dependent boundary parameters, and a first-order transition occurs at q^2 = 1 + Δ.
  • Several particle classes: Ferrari and Martin’s construction generates steady states for multi-class particle systems and can provide operators and auxiliaries for more than three classes.This generalizes an earlier construction for the two-class problem.

10. Discrete-time updating schemes

The review develops matrix product descriptions for sublattice, ordered sequential, and fully parallel ASEP dynamics. These schemes differ in update organization and correlation formulas, while sharing phase-diagram structures in important cases.

  • Transfer-matrix framework: Discrete-time steady-state weights are eigenvectors with eigenvalue one of the dynamics’ transfer matrices.The transfer matrix sums predecessor configuration weights multiplied by one-step transition probabilities.
  • Sublattice updating: Sublattice updating splits each timestep into simultaneous even-bond and odd-bond updates, requiring an even number of sites.Different hatted and unhatted matrices represent even and odd sublattices in the matrix product state.
  • Sublattice updating: The sublattice cancellation mechanism moves hatted matrices between sublattices and produces algebraic relations whose solutions can reuse the random-sequential ASEP matrices.An ansatz shifts the hatted matrices by ±λ times the identity, reducing the relations to the usual PASEP algebra with redefined boundary rates.
  • Ordered updating: Backward-ordered updating has the same phase diagram as sublattice updating, although its exact correlation functions depend on update details; forward ordering follows by particle-hole symmetry.The auxiliary hatted matrices appear during the update and move through the lattice rather than remaining in the final weights.
  • Fully parallel updating: Fully parallel dynamics updates all bonds and boundary sites simultaneously and is particularly natural for traffic-flow modeling.Its phase diagram resembles the continuous-time case, but the transition lines to maximum current occur at α = 1 −√(1 − p) and β = 1 −√(1 − p).

11. Summary and outstanding challenges

The review surveys the physical phenomena and model classes accessible through matrix product methods, then identifies important systems and update schemes that remain unresolved. It concludes that systematic construction from microscopic dynamics remains a remote goal.

  • Summary: Matrix product methods describe driven systems with phase transitions induced by boundary interactions or by differing dynamics among particle species.The reviewed models include hard-core diffusing particles, multi-species extensions, and some non-conserving reactions.
  • Outstanding challenges: A systematic procedure for constructing any nonequilibrium steady state in matrix product form from microscopic dynamics remains a remote goal.Existence proofs indicate that formulations should nearly always be possible, but practical work still relies on particular examples.
  • Outstanding challenges: Several simple systems with nontrivial nonequilibrium behavior remain unsolved by matrix products or other exact methods.These systems are presented as challenges for future research.
  • Driven n-mers with open boundaries: Open-boundary exclusion with extended particles lacks a matrix product solution because boundary cancellation is difficult and suitable boundary conditions are unclear.The review’s solved models use particles occupying a single lattice site.
  • Outstanding challenges: The ABC model is exactly solvable for equal species numbers but remains incompletely solved more generally, despite phase separation when q ≠ 1.Equal numbers permit detailed balance and are among the few cases with a known free-energy functional.
  • Outstanding challenges: Matrix product steady states for arbitrary fixed or shuffled sequential update orders remain open problems.Shuffled updating updates each site or bond exactly once per timestep and has been argued relevant to pedestrian dynamics.

A. Method of characteristics

The method of characteristics solves first-order density equations by identifying curves along which boundary or initial information propagates. For nonlinear transport, these characteristics can generate richer boundary-induced behavior, including stationary shocks.

  • Characteristic construction: The method of characteristics identifies curves along which information from boundary or initial conditions propagates through the space-time domain.The characteristic curves satisfy differential conditions derived from the density equation.
  • Characteristic construction: The solution can be represented by two families of curves whose constants are fixed by characteristic relations and initial data.The functions φ and ψ remain constant along the respective characteristic families, while F is fixed by the initial condition.
  • Characteristic behavior: For c = 0, characteristics are straight lines with slope v_g(ρ), and the density remains constant along each line.For more general cases, characteristics curve and the density varies along them.
  • Characteristic behavior: More complicated characteristic curves can produce richer boundary-induced phase transitions, including stationary shocks.The density evolution of an initial profile patch determines the characteristic trajectory.

B. Generating functions and asymptotics

This appendix develops coefficient-extraction and asymptotic tools for generating functions, including Lagrange inversion and singularity analysis. These methods identify coefficients exactly or asymptotically from functional relations and nearby singularities.

  • Generating functions: The appendix defines coefficient notation for extracting the coefficient of x^n from a formal power series.It uses {x^n}f(x) for this coefficient.
  • Generating functions: Lagrange inversion converts a functional relation for f into coefficients of an arbitrary function F(f).The procedure first inverts the relation expressing x as a function of f.
  • Asymptotics: A singularity at x_0 contributes an asymptotic term involving x_0^-n and a power n^-µ to the coefficient sequence.The constants A and µ are determined from the singular behavior.
  • Asymptotics: For sufficiently large n, the singularity closest to the origin dominates the coefficient asymptotics.Asymptotic equivalence is introduced to formalize the comparison of sequences.
  • Asymptotics: The treatment covers both poles and algebraic singularities by decomposing the generating function into regular and singular parts.Gamma-function expressions extend the coefficient formulas from integer poles to noninteger algebraic singularities.

C. Equivalence of the integral and sum representations of the ASEP normalisation

The appendix proves that two representations of the ASEP normalisation are equivalent by showing that their generating functions coincide. The integral is converted into a contour integral and evaluated by residues.

  • Generating-function comparison: The equivalence proof compares the generating functions obtained from the finite-sum and integral representations of Z_N.The finite-sum expression produces a generating function that is developed further.
  • Contour representation: The integral representation is recast as a contour integral using the change of variable u = e^iθ.The contour is the positively oriented unit circle in the complex u plane.
  • Contour representation: For |z| < 1/4, summing a geometric series yields the generating function used in the contour calculation.The convergence condition is stated explicitly for the geometric-series step.
  • Residue evaluation: With a and b < 1, the unit-circle contour encloses simple poles at u = a, u = b, and u = u−.The contour integral is evaluated using the residue theorem.
  • Residue evaluation: The residues from the three poles produce an expression algebraically equivalent to the generating-function result from the finite sum.The final algebra establishes agreement with equation (C.2).

D. Matrix representations

This appendix collects representations of the matrix product algebras developed throughout the work. It serves as a reference for the algebraic constructions used in the paper.

  • Matrix representations: The appendix assembles several representations of the matrix product algebras discussed in the paper.It is presented as a consolidated reference.
  • Matrix representations: The collected representations provide algebraic forms associated with the paper’s matrix product treatment.The passage identifies their relationship to the preceding work.
  • Matrix representations: The appendix’s purpose is organizational: it gathers previously discussed matrix product algebras in one place.No additional representation-specific result is stated in the passage.

D.1. Representations of PASEP algebra

The appendix reviews multiple representations of the PASEP algebra, including alternatives for boundary injection and extraction. It also identifies finite-dimensional parameter curves and discusses q → 1 limits.

  • Representations of PASEP algebra: The appendix begins with representations of the PASEP algebra for injection at the left boundary and extraction at the right boundary.These representations are recapped before introducing alternatives.
  • Finite-dimensional representations: For certain parameter curves, the representations become finite dimensional because c_n = 0 disconnects the upper-left matrix block from the rest.Such curves were noted previously in the more general γ, δ ≠ 0 case, and finite-dimensional representations were catalogued there.
  • Limits: The representations associated with equations (5.12) have well-defined limits as q → 1.This contrasts with the representation whose q → 1 limit is undefined.
  • Representations of PASEP algebra: Three PASEP representations are presented, including a second representation from Section 5.3 and a third not previously encountered.The representations generalise three forms first given for the totally asymmetric case.
  • Representations of PASEP algebra: The second representation uses parameters a and b together with a normalization constant κ usually chosen so that ⟨V|W⟩ = 1.Its q → 1 limit is not defined.

D.2. Representations of two-species and multispecies algebras

This section gives matrix representations for physically relevant two-species algebras and their multispecies generalisations, including several classified solution classes. It presents projector, disordered-rate, and deformed-commutator constructions, alongside explicit representations and a tracelessness property.

  • Representations are constructed for the nontrivial algebras classified earlier and for their multispecies generalisations.
  • Solution AII and multispecies generalisation: Solutions AII and its multispecies generalisation use matrices E and D(v), with a representation involving β(v) and powers of v.
  • Solution BI and BII: Solutions BI and BII are specified through the algebra containing relations among X0, X1, and X2 with parameters p1, p2, q1, and q2.
  • Solution BI and BII: For solution BII, X2 can be represented as the projector |V⟩⟨W| using a presentation of the PASEP algebra.
  • Solution CII and multispecies generalisation: Solution CII treats the ASEP with disordered hopping rates and gives its matrix algebra and multispecies generalisation.
  • Solution D and Deformed commutators: Solution D operators can be built from tensor products of operators obeying deformed commutator relations, with r = 1 recovering the usual commutator.The deformed commutator appears in PASEP analysis and some two-species models; X1 and every product containing at least one X1 are traceless.
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