[Paper Review] Absorbing State Phase Transition in presence of Conserved Continuous Local Field

Mahashweta Basu, Ujjal Gayen+1 more|arXiv (Cornell University)|Feb 8, 2011

Theoretical and Computational Physics1 references3 citations

TL;DR

This paper studies a one-dimensional stochastic lattice model with a conserved continuous local field (energy) where activity is triggered when site energy exceeds a threshold. The system exhibits a continuous absorbing state phase transition at a critical energy density, with critical exponents distinct from directed percolation, establishing a new universality class for conserved continuous field models with uncountably infinite absorbing states.

ABSTRACT

We study absorbing state phase transition in one dimension in presence of a conserved continuous local field (CCLF) called energy. A pair of sites on a lattice is said to be active if one or both sites posses more energy than a pre-defined threshold. The active pair of sites are allowed to redistribute their energy following a stochastic rule. We show that, the CCLF model undergo a continuous absorbing state transition when energy per site is decreased below a critical value. The critical exponents are found to be different from those of DP.

Motivation & Objective

• To investigate absorbing state phase transitions in systems with a conserved continuous local field (CCLF), such as energy, in one dimension.
• To determine whether the presence of a continuous field and uncountably infinite absorbing states alters critical behavior compared to standard directed percolation (DP) universality.
• To examine the robustness of critical exponents under variations of the model's dynamics and parameters.
• To assess whether the system belongs to a new universality class distinct from known models like CTTP or DP.
• To establish a critical point and scaling behavior for the order parameter and relaxation dynamics in finite and infinite systems.

Proposed method

• Define a one-dimensional lattice with periodic boundary conditions, where each site holds a continuous energy variable $ E_i $, conserved globally.
• Define an active pair as two neighboring sites where at least one has energy $ E_i > w $, with $ w $ being a threshold; only active pairs undergo energy exchange.
• Implement a stochastic, energy-conserving update rule: $ E_i o ho E_i + u (E_i + E_j) $, $ E_j o (1- ho) E_j + u (E_i + E_j) $, with $ ho o ext{uniform}(0,1) $, ensuring total energy conservation.
• Measure the order parameter $ ho(t,e) $, defined as the fraction of active pairs at time $ t $, to probe the transition from active to absorbing state.
• Use finite-size scaling and data collapse techniques to extract critical exponents $ eta $, $ u_ot $, $ u_ op $, $ heta $, and $ au $, and test scaling forms.
• Apply iterative fitting using $ ho(t,e_c) = ho(t,e) / (1 + B(e - e_c) t^{1/ u_ op}) $ to refine the critical energy $ e_c $ and confirm power-law decay.

Experimental results

Research questions

• RQ1 Does the presence of a conserved continuous local field (energy) in a one-dimensional system lead to a continuous absorbing state phase transition?
• RQ2 Are the critical exponents of this transition different from those of the directed percolation (DP) universality class?
• RQ3 How does the system's behavior change with system size, and does the critical energy $ e_c $ converge in the thermodynamic limit?
• RQ4 Can the model be described by a scaling function consistent with known universality classes, or does it define a new one?
• RQ5 What is the role of uncountably infinite absorbing states (due to continuous energy) in altering critical behavior compared to discrete models?

Key findings

• The model exhibits a continuous absorbing state phase transition at a critical energy density $ e_c = 0.7508 $ in the thermodynamic limit, with $ e_c = 0.7503 $ for $ L = 1024 $.
• The critical exponent $ u_ op = 2.6(4) $ is obtained from data collapse and linear fits of $ A^{-1} ho(t,e)t^eta - 1 $, consistent with $ 1/ u_ op = 0.379 $.
• The order parameter decay exponent is $ eta = 0.19 $, with $ ho(t,e) o t^{-eta} $ at criticality, and the scaling function $ ho(t,e) o At^{-eta}(1 + B(e - e_c)t^{1/ u_ op}) $ holds for $ t o ext{small} $.
• The critical exponent $ eta $, derived from $ ho_s o (e - e_c)^eta $, is consistent across system sizes and confirms the transition is continuous.
• The critical point $ e_c(L) $ varies linearly with $ 1/L $, and extrapolation to $ L o ty $ yields $ e_c = 0.7508 $, indicating convergence and absence of strong finite-size effects.
• The model's critical exponents differ significantly from those of directed percolation and the conserved threshold transfer process (CTTP), establishing a new universality class for systems with conserved continuous fields and uncountably infinite absorbing states.