[Paper Review] Scaling of many-particle correlations in a dissipative sandpile

Николай Михайлович Боголюбов, A. G. Pronko+1 more|arXiv (Cornell University)|Feb 28, 2011

Theoretical and Computational Physics26 references3 citations

TL;DR

This paper derives an exact solution for the conditional probability that K grains preserve their initial order during an avalanche in a 2D directed dissipative sandpile, mapping the problem to a (1+1)D system with discrete time. The solution reveals non-trivial scaling with a power-law decay whose exponent depends nonlinearly on K, and establishes exact equivalences to the 'random-turns' vicious walker model and the Heisenberg XX spin chain correlation function, linking sandpile dynamics to free fermion systems via the Jordan-Wigner transformation.

ABSTRACT

The two dimensional directed sandpile with dissipation is transformed into a (1+1) dimensional problem with discrete space and continuous `time'. The master equation for the conditional probability that K grains preserve their initial order during an avalanche can thereby be solved exactly, and an explicit expression is given for the asymptotic form of the solution for an infinite as well as for a semi-infinite lattice in the horizontal direction. Non-trivial scaling is found in both cases. This conditional probability of the sandpile model is shown to be equal to a K-spin correlation function of the Heisenberg XX spin chain, and the sandpile problem is also shown to be equivalent to the `random-turns' version of vicious walkers.

Motivation & Objective

• To understand the scaling behavior of many-particle correlations in a dissipative sandpile model.
• To derive an exact analytic expression for the conditional probability that K grains maintain their initial order during an avalanche.
• To establish connections between the sandpile model, the 'random-turns' vicious walker problem, and the Heisenberg XX spin chain.
• To analyze how non-zero dissipation modifies the scaling behavior, particularly through an exponential cutoff and nonlinear K-dependent scaling exponent.
• To demonstrate the equivalence of the sandpile conditional probability to a many-spin correlation function in the XX spin chain, thereby linking it to free fermion systems.

Proposed method

• Transform the 2D directed sandpile with dissipation into a (1+1)D problem with discrete time and discrete spatial lattice.
• Solve the master equation for the conditional probability that K grains preserve their initial order during an avalanche exactly.
• Use saddle-point approximation and asymptotic analysis to derive the large-time (n→∞) behavior of the discrete multi-grain probability.
• Establish equivalence between the sandpile conditional probability and the partition function of K 'random-turns' vicious walkers via generating function techniques.
• Prove that the same conditional probability equals a K-spin correlation function in the Heisenberg XX spin chain using the Jordan-Wigner transformation.
• Analyze both infinite and semi-infinite lattice geometries to compare boundary condition effects on scaling exponents.

Experimental results

Research questions

• RQ1 How do many-particle correlations scale in a dissipative sandpile model?
• RQ2 What is the exact form of the conditional probability that K grains maintain their initial order during an avalanche?
• RQ3 How does non-zero dissipation affect the scaling behavior of the conditional probability?
• RQ4 What is the connection between the sandpile model and the 'random-turns' vicious walker problem?
• RQ5 How is the sandpile conditional probability related to the Heisenberg XX spin chain correlation function?

Key findings

• The conditional probability for K grains to preserve their initial order scales as a power law ∼n^−γ in the large-time limit, with a scaling exponent γ that depends nonlinearly on K.
• For the infinite lattice, the scaling exponent γ is given by the expression derived in equation (3.13), while for the semi-infinite lattice it is given by equation (3.24), showing distinct boundary effects.
• Non-zero dissipation introduces an exponential cutoff in the asymptotic form of the conditional probability, modifying the power-law decay.
• The conditional probability is exactly equal to the partition function of K 'random-turns' vicious walkers, establishing a direct link to this well-known stochastic model.
• The same conditional probability is identified as a K-spin correlation function in the Heisenberg XX spin chain, implying a deep connection to free fermion systems via the Jordan-Wigner transformation.
• The discrete multi-grain probability in the large-n limit scales identically to the continuous-time counterpart, with the replacement t→n/K, confirming consistency across time discretization.