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Register a new userFinite-size scaling study of dynamic critical phenomena in a vapor-liquid transition
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Via a combination of molecular dynamics (MD) simulations and finite-size scaling (FSS) analysis, we study dynamic critical phenomena for the vapor-liquid transition in a three dimensional Lennard-Jones system. The phase behavior of the model, including the critical point, have been obtained via the Monte Carlo simulations. The transport properties, viz., the bulk viscosity and the thermal conductivity, are calculated via the Green-Kubo relations, by taking inputs from the MD simulations in the microcanonical ensemble. The critical singularities of these quantities are estimated via the FSS method. The results thus obtained are in nice agreement with the predictions of the dynamic renormalization group and mode-coupling theories.
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We test an improved finite-size scaling method for reliably extracting the critical temperature $T_{rm BKT}$ of a Berezinskii-Kosterlitz-Thouless (BKT) transition. Using known single-parameter logarithmic corrections to the spin stiffness $rho_s$ at $T_{rm BKT}$ in combination with the Kosterlitz-Nelson relation between the transition temperature and the stiffness, $rho_s(T_{rm BKT})=2T_{rm BKT}/pi$, we define a size dependent transition temperature $T_{rm BKT}(L_1,L_2)$ based on a pair of system sizes $L_1,L_2$, e.g., $L_2=2L_1$. We use Monte Carlo data for the standard two-dimensional classical XY model to demonstrate that this quantity is well behaved and can be reliably extrapolated to the thermodynamic limit using the next expected logarithmic correction beyond the ones included in defining $T_{rm BKT}(L_1,L_2)$. For the Monte Carlo calculations we use GPU (graphical processing unit) computing to obtain high-precision data for $L$ up to 512. We find that the sub-leading logarithmic corrections have significant effects on the extrapolation. Our result $T_{rm BKT}=0.8935(1)$ is several error bars above the previously best estimates of the transition temperature; $T_{rm BKT} approx 0.8929$. If only the leading log-correction is used, the result is, however, consistent with the lower value, suggesting that previous works have underestimated $T_{rm BKT}$ because of neglect of sub-leading logarithms. Our method is easy to implement in practice and should be applicable to generic BKT transitions.
A smooth cut-off formulation of the Hierarchical Reference Theory (HRT) is developed and applied to a Yukawa fluid. The HRT equations are derived and numerically solved leading to: the expected renormalization group structure in the critical region, non classical critical exponents and scaling laws, a convex free energy in the whole phase diagram (including the two-phase region), finite compressibility at coexistence, together with a fully satisfactory comparison with available numerical simulations. This theory, which also guarantees the correct short range behavior of two body correlations, represents a major improvement over the existing liquid state theories.
A two parameter percolation model with nucleation and growth of finite clusters is developed taking the initial seed concentration rho and a growth parameter g as two tunable parameters. Percolation transition is determined by the final static configuration of spanning clusters. A finite size scaling theory for such transition is developed and numerically verified. The scaling functions are found to depend on both g and rho. The singularities at the critical growth probability gc of a given rho are described by appropriate critical exponents. The values of the critical exponents are found to be same as that of the original percolation at all values of rho at the respective gc . The model then belongs to the same universality class of percolation for the whole range of rho.
We investigate the use of matrix product states (MPS) to approximate ground states of critical quantum spin chains with periodic boundary conditions (PBC). We identify two regimes in the (N,D) parameter plane, where N is the size of the spin chain and D is the dimension of the MPS matrices. In the first regime MPS can be used to perform finite size scaling (FSS). In the complementary regime the MPS simulations show instead the clear signature of finite entanglement scaling (FES). In the thermodynamic limit (or large N limit), only MPS in the FSS regime maintain a finite overlap with the exact ground state. This observation has implications on how to correctly perform FSS with MPS, as well as on the performance of recent MPS algorithms for systems with PBC. It also gives clear evidence that critical models can actually be simulated very well with MPS by using the right scaling relations; in the appendix, we give an alternative derivation of the result of Pollmann et al. [Phys. Rev. Lett. 102, 255701 (2009)] relating the bond dimension of the MPS to an effective correlation length.
We develop a scaling theory for the finite-size critical behavior of the microcanonical entropy (density of states) of a system with a critically-divergent heat capacity. The link between the microcanonical entropy and the canonical energy distribution is exploited to establish the former, and corroborate its predicted scaling form, in the case of the 3d Ising universality class. We show that the scaling behavior emerges clearly when one accounts for the effects of the negative background constant contribution to the canonical critical specific heat. We show that this same constant plays a significant role in determining the observed differences between the canonical and microcanonical specific heats of systems of finite size, in the critical region.
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