Periodic TASEP: relaxation gaps, not energies¶

See the worked relaxation and scaling tutorial for plots, downloadable exports and a comparison of fixed density with one particle.

Status: precision-generic library and bethe-tasep-pbc frontend implemented. This is the totally asymmetric member of the ASEP family. It is not yet a solver for bidirectional hopping, open reservoirs, or a complete non-Hermitian spectrum. Bidirectional hopping is available in the separate ASEP library/frontend.

Physical convention¶

On a ring of L sites, each of N hard-core particles jumps from j to j+1 at rate r>0 if the destination is empty. The probability vector obeys dP/dt=M P: columns of M sum to zero, off-diagonal entries are transition rates, and diagonal entries are minus total escape rates. These are inverse time units, not quantum energies.

The stationary eigenvalue is zero. The nonzero eigenvalues have negative real parts. We report a leading eigenvalue lambda with nonnegative imaginary part, and the positive relaxation gap g=-Re(lambda). Its conjugate is also present; Im(lambda) describes oscillation, not a second contribution to the decay rate. The relaxation time is 1/g. N=0 or L has just one configuration and no nonzero relaxation mode, so its gap is absent, not a claimed zero gap.

Command-line use¶

bethe-tasep-pbc 32 --particles 8 --roots
bethe-tasep-pbc 64 --particles 32 --rate 2 --precision long-double
bethe-tasep-pbc 32 --particles 8 --roots --json rates.json \
  --csv-table relaxation=rates.csv --tsv-table roots=roots.tsv
bethe-tasep-pbc --references

L and --particles are required. --rate defaults to one; the shared --precision choices are fp64, long-double and fp128 when MPLAPACK is enabled. With Uni20 MPFR scalars enabled, bits-N and digits-N select explicit arbitrary precision, using normal CPU/provider LU dispatch. The relaxation table has one row, with lambda_real, lambda_imag, gap, frequency, residual, iteration counts, convergence and status. Frequency is the nonnegative angular frequency, equal to lambda_imag, in inverse time—not cycles per time. gap=-lambda_real; the conjugate partner is implicit. There are no energy or momentum columns.

has_mode=false distinguishes the successful empty/full sector, whose mode observables are null, from numerical failure in a nontrivial sector. The exact stationary eigenvalue zero is recorded in metadata independently of the gap calculation. --roots adds reduced-population fugacities Z_real, Z_imag; these are not the lattice wave numbers z or necessarily the user's N particles.

Metadata includes the generator/sign convention, rate units, root reduction, precision, all work budgets, build/invocation provenance and CPU time. All shared output options, including streaming and no-retain exports, apply. Ordinary help/calculations omit the bibliography; use --references.

--max-iterations, --max-seed-iterations and --max-sites expose the library budgets below. Numerical failures exit 2 with null JSON observables/root coordinates (empty CSV/TSV fields). Invalid inputs exit 1 before opening files, including existing --force targets. --left-rate and --excitations are not supported: this command is a TASEP relaxation-gap calculation, not a spectrum scan.

Library API and output contract¶

#include <bethe/tasep.hpp>
auto state = bethe::tasep::relaxation_gap(32, 8, 1.0);
if (state.converged && state.gap) {
  // *state.eigenvalue is complex, *state.gap is positive.
}

All arithmetic stays in the selected native type or explicit MPFR precision. Particle–hole symmetry reduces the root count to effective_particles=min(N,L-N). The retained roots therefore describe this reduced filling, not necessarily the user's particle coordinates. Choosing the nonnegative-frequency representative fixes a conjugate pair convention, not an independently assigned physical momentum.

Options<Real> supplies a scaled equation tolerance (default 128 working-precision epsilons), max_seed_iterations (1000 all-root sweeps), max_iterations (100 coupled Newton updates), and a max_sites work budget (256). Both iteration budgets are explicit; there are no hidden retries. One-particle/one-hole rates are analytic and need no iterations. stationary_only is a successful empty/full sector with no eigenvalue or gap; converged supplies both. Failures (seed_limit, iteration_limit, stalled, precision_limit) never publish either observable. Any retained roots are diagnostics only. Invalid inputs and exceeding the size budget throw.

Equations and branch selection¶

For unit rate and reduced population n, use the fugacity coordinate Z=2/z-1:

\[ \begin{aligned} (1-Z_j)^n(1+Z_j)^{L-n}&=Y,\\{} Y&=-2^L\prod_j\frac{Z_j-1}{Z_j+1},\\{} \lambda&=\sum_j\frac{Z_j-1}{2}. \end{aligned} \]

These are equations (2)–(8) of Golinelli–Mallick (2005). Their first-relaxation branch exchanges the upper roots on either side of the Cassini pinch relative to the n roots with largest real parts, equations (12)–(14). We use that finite-size branch, not their asymptotic large-L gap as an approximation. Gwa–Spohn (1992) is the earlier spectral-gap reference.

The initial Y is negative, with \(\log \lvert Y \rvert =L \log (2)+n \log (n /L)+(L -n) \log (1-n /L)+3.5\). All L polynomial roots are recovered by Ehrlich–Aberth iteration using the factored logarithmic expression. Expanding this polynomial in monomials can destroy the branch selection at larger half-filled sizes even when the final Bethe residual looks small. Root separation and factored residuals are checked before the literature branch is selected. The ratio is inverted when necessary to keep its exponential bounded; this avoids seed-iteration overflow in dilute large rings without relaxing the residual check.

The final solve treats selected complex roots and log(Y) together, with analytic derivatives and real/imaginary components fed to the shared recoverable pivoted Newton solver. Imaginary residuals are reduced modulo 2*pi. The reported norm is the largest complex logarithmic residual divided by L. Distinct roots and the first-harmonic translation factor are checked before publishing the rate. Residual convergence is a numerical check, not an interval certification or a completeness proof. Larger user-selected budgets can be expensive and do not guarantee convergence or uniform relative gap accuracy.

Validation and next steps¶

Tests assemble the full physical Markov matrix independently and compare its leading nonzero eigenvalues for every filling at L=2 through 10. A separate 90-digit ten-configuration matrix reference tests the native precision at L=5, N=2. All enabled precisions check the original multiplicative equations, analytic Jacobian, rate scaling, failure budgets and selected rings through 256 sites. Dense Markov diagonalization is test-only, never the solver backend.

The frontend also tests all precision modes against the independent five-site reference, rate scaling, exact one-particle/one-hole limits, empty/full sectors, budget/range failures, exports, citation policy and invalid-input file protection.

The ASEP library/frontend uses this TASEP seed with its own bidirectional equations and branch continuation.

Independent sizes and fillings compose through the existing owning Async grid; the TASEP/XYZ example demonstrates a filling curve with bounded admission and complete diagnostics. Each Cassini seed and Newton solve remains sequential. A positive rate change only scales the final eigenvalue and gap, so a rate-only curve should reuse one unit-rate state. The single-request CLI stays serial. Mixed-precision requests own their rates and options until their final Async result completes.

Explicit precision¶

bethe-tasep-pbc 32 --particles 8 --precision bits-160 --roots

using Real = uni20::mpreal;
auto p = uni20::Precision::bits(160);
bethe::tasep::Options<Real> options(p);
auto state = bethe::tasep::relaxation_gap(32, 8, Real(1, p), options);

Rate, tolerance, complex roots, seed constants, real-block Newton solves and failure diagnostics retain the requested precision. Inputs must match the options' finite precision; no ambient MPFR precision is changed. Independent 90-digit Markov-matrix eigenvalues demonstrate accuracy growth at 80, 160 and 256 bits. Mixed-precision owning Async tests include successful and failed requests, and compare every result field with direct evaluation.

The independent matrix reference can be regenerated with scripts/reference_tasep_precision.py (requires mpmath). It assembles the physical ten-configuration generator and diagonalizes it at 100 and 130 decimal digits, without using Bethe roots.

The seed and Newton budgets retain their native defaults. More bits do not guarantee convergence within these budgets or a better state-family selection.