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arxiv: 2604.25060 · v1 · submitted 2026-04-27 · ❄️ cond-mat.stat-mech · physics.chem-ph

Statistical mechanics in continuous space with tensor network methods

Gunhee Park , Tomislav Begu\v{s}i\'c , Si-Jing Du , Johnnie Gray , Garnet Kin-Lic Chan This is my paper

Pith reviewed 2026-05-07 17:31 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.chem-ph
keywords tensor networksstatistical mechanicscontinuous spacehard-disk modelpartition functioncoarse-grainingboundary contractionthermodynamics
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The pith

Tensor networks compute thermodynamic quantities for continuous-space particles by mapping them to locality-preserving lattice models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to extend tensor network methods, traditionally restricted to lattice systems, to interacting particles moving in continuous space. It does so by discretizing the space into a grid and applying cell-based coarse-graining to produce an effective lattice model whose interactions remain local. The partition function of this lattice model is encoded as a tensor network, and thermodynamic observables are extracted by contracting the network from the boundary inward. The framework is applied to the two-dimensional hard-disk fluid, where the resulting calculations are compared against Monte Carlo simulations. If the mapping is accurate, tensor networks become available for a wide range of classical statistical-mechanics problems that were previously inaccessible to them.

Core claim

Through a real-space discretization combined with a cell-based coarse-graining scheme, we formulate an effective lattice model that explicitly preserves spatial locality. The partition function of this model is represented as a tensor network, and the thermodynamic quantities are computed via boundary contraction. We apply this framework to the two-dimensional hard-disk problem and demonstrate the strengths of the tensor-network formulation compared to existing Monte Carlo simulations.

What carries the argument

Real-space discretization combined with cell-based coarse-graining that yields a locality-preserving effective lattice model whose partition function is encoded as a tensor network and evaluated by boundary contraction.

Load-bearing premise

The discretization and cell-based coarse-graining must produce an effective lattice model whose thermodynamics match those of the original continuous-space system without uncontrolled errors that grow with size or density.

What would settle it

Systematic deviation between tensor-network results and converged Monte Carlo or exact results for pressure or free energy of the hard-disk system as density or system size is increased.

Figures

Figures reproduced from arXiv: 2604.25060 by Garnet Kin-Lic Chan, Gunhee Park, Johnnie Gray, Si-Jing Du, Tomislav Begu\v{s}i\'c.

Figure 2
Figure 2. Figure 2: FIG. 2. (a) Diagrammatic representation of the factor graph view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. (a) Particles in a box. (b) Same system in the site rep view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. 2 view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Within the infinite TN framework, the density is plot view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Pair correlation functions computed from infinite TN and MC calculations. For the MC reference, the GCMC data view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Free energy convergence using the TN boundary view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Density as a function of lattice discretization constant view at source ↗
read the original abstract

Tensor network (TN) methods are well established for computing partition functions in statistical mechanics, though this use has traditionally been limited to lattice models. We extend the scope of TN methodology to interacting particle systems in continuous space. Through a real-space discretization combined with a cell-based coarse-graining scheme, we formulate an effective lattice model that explicitly preserves spatial locality. The partition function of this model is represented as a TN, and the thermodynamic quantities are computed via boundary contraction. We apply this framework to the two-dimensional hard-disk problem and demonstrate the strengths of the TN formulation compared to existing Monte Carlo simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims to extend tensor network methods, traditionally limited to lattice models, to interacting particle systems in continuous space. It does so by combining real-space discretization with a cell-based coarse-graining scheme to produce an effective local lattice model whose partition function is encoded as a TN and evaluated by boundary contraction. The framework is applied to the two-dimensional hard-disk system, where the authors assert that it demonstrates advantages relative to existing Monte Carlo simulations.

Significance. If the discretization and coarse-graining map the continuous-space partition function to a lattice model whose thermodynamics converge to the exact continuous result in the appropriate limit, the work would meaningfully enlarge the domain of TN techniques to off-lattice problems. The explicit retention of spatial locality and the use of boundary contraction are technically attractive features that could complement Monte Carlo sampling in regimes with strong correlations or slow mixing.

major comments (1)
  1. [Abstract and method description] The central claim that the effective lattice model faithfully reproduces the thermodynamics of the continuous hard-disk system rests on an unverified assertion. The abstract states that the method works for hard disks and shows strengths versus Monte Carlo, but provides no quantitative error analysis, convergence data with respect to cell size or discretization parameters, or systematic extrapolation to the continuous limit. Without such controls, residual approximation errors that fail to vanish with system size or density would make the TN contraction compute the wrong model, rendering any claimed advantage over Monte Carlo irrelevant.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need for explicit verification of the continuous-space limit. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and method description] The central claim that the effective lattice model faithfully reproduces the thermodynamics of the continuous hard-disk system rests on an unverified assertion. The abstract states that the method works for hard disks and shows strengths versus Monte Carlo, but provides no quantitative error analysis, convergence data with respect to cell size or discretization parameters, or systematic extrapolation to the continuous limit. Without such controls, residual approximation errors that fail to vanish with system size or density would make the TN contraction compute the wrong model, rendering any claimed advantage over Monte Carlo irrelevant.

    Authors: We agree that a systematic quantitative study of the discretization and coarse-graining errors is required to rigorously establish convergence to the continuous hard-disk thermodynamics. The present manuscript reports thermodynamic quantities obtained at fixed discretization parameters and compares them to Monte Carlo benchmarks, but does not contain a dedicated convergence analysis with respect to cell size or an extrapolation procedure. We will add this analysis to the revised version, including explicit error estimates and scaling plots that demonstrate the vanishing of the approximation error in the appropriate limit. This addition will directly address the concern that the TN contraction might be solving an incorrect model. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper maps continuous-space hard-disk systems to an effective lattice model via explicit real-space discretization and cell-based coarse-graining, then represents the partition function as a tensor network and computes quantities by boundary contraction. No load-bearing step reduces computed thermodynamic quantities to fitted parameters, self-referential definitions, or self-citation chains that presuppose the target result. The central claim is a methodological construction whose validity is checked against independent Monte Carlo benchmarks rather than by construction. Any self-citations to prior TN work are non-load-bearing and do not substitute for the discretization step itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unproven premise that the chosen discretization and coarse-graining preserve the continuous-space partition function to sufficient accuracy; no free parameters or new entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Real-space discretization combined with cell-based coarse-graining yields an effective lattice model that preserves spatial locality and the thermodynamics of the original continuous system.
    Invoked when the authors state that the effective lattice model is formulated and its partition function is represented as a TN.

pith-pipeline@v0.9.0 · 5410 in / 1248 out tokens · 59048 ms · 2026-05-07T17:31:40.790097+00:00 · methodology

discussion (0)

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Reference graph

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    (b) The number of Monte Carlo steps in the Wang-Landau algorithm as a function of area

    Free energy density values for different system sizes are plotted in the inset. (b) The number of Monte Carlo steps in the Wang-Landau algorithm as a function of area. The inset shows a comparison of the free energy density calculated via the TN and MC methods. See the main text for the details of the Monte Carlo procedures. a versatile tool for studying ...

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