Machta, Jonathan
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Professor, Department of Physics
Last Name
Machta
First Name
Jonathan
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Physics
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Introduction
My research is in the area of theoretical condensed matter and statistical physics. My current research involves theoretical and computational studies of spin systems and applications of computational complexity theory to statistical physics.
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Publication Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity(2000-01-01) Moore, C; Machta, JThe computational complexity of internal diffusion-limited aggregation (DLA) is examined from both a theoretical and a practical point of view. We show that for two or more dimensions, the problem of predicting the cluster from a given set of paths is complete for the complexity class CC, the subset of P characterized by circuits composed of comparator gates. CC-completeness is believed to imply that, in the worst case, growing a cluster of size n requires polynomial time in n even on a parallel computer. A parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths, and then corrects defects in the guessed cluster through a nonlocal annihilation process. The parallel running time of the relaxation algorithm for two-dimensional internal DLA is studied by simulating it on a serial computer. The numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n. Thus the computational resources needed to grow large clusters are significantly less on average than the worst-case analysis would suggest. For a parallel machine with k processors, we show that random clusters in d dimensions can be generated in $$\mathcal{O}$$ ((n/k+logk)n 2/d ) steps. This is a significant speedup over explicit sequential simulation, which takes $$\mathcal{O}$$ (n 1+2/d ) time on average. Finally, we show that in one dimension internal DLA can be predicted in $$\mathcal{O}$$ (logn) parallel time, and so is in the complexity class NC.Publication Critical Behavior of the Chayes–Machta–Swendsen–Wang Dynamics(2007-01-01) Deng, Y; Garoni, T; Machta, J; Ossola, G; Polin, M; Sokal, AWe study the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts model to noninteger q, in two and three spatial dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z≥α/ν is close to but probably not sharp in d=2 and is far from sharp in d=3, for all q. The conjecture z≥β/ν is false (for some values of q) in both d=2 and d=3.Publication Overcoming the slowing down of flat-histogram Monte Carlo simulations: Cluster updates and optimized broad-histogram ensembles(2005-01-01) Wu, Y; Korner, M; Colonna-Romano, L; Trebst, S; Gould, H; Machta, Joonathan; Troyer, MWe study the performance of Monte Carlo simulations that sample a broad histogram in energy by determining the mean first-passage time to span the entire energy space of d-dimensional ferromagnetic Ising/Potts models. We first show that flat-histogram Monte Carlo methods with single-spin flip updates such as the Wang-Landau algorithm or the multicanonical method perform suboptimally in comparison to an unbiased Markovian random walk in energy space. For the d=1, 2, 3 Ising model, the mean first-passage time τ scales with the number of spins N=Ld as τ∝N2Lz. The exponent z is found to decrease as the dimensionality d is increased. In the mean-field limit of infinite dimensions we find that z vanishes up to logarithmic corrections. We then demonstrate how the slowdown characterized by z>0 for finite d can be overcome by two complementary approaches—cluster dynamics in connection with Wang-Landau sampling and the recently developed ensemble optimization technique. Both approaches are found to improve the random walk in energy space so that τ∝N2 up to logarithmic corrections for the d=1, 2 Ising model.Publication Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points via Ginzburg–Landau Polynomials(2008-01-01) Ellis, R; Machta, J; Otto, PThe purpose of this paper is to prove connections among the asymptotic behavior of the magnetization, the structure of the phase transitions, and a class of polynomials that we call the Ginzburg–Landau polynomials. The model under study is a mean-field version of a lattice spin model due to Blume and Capel. It is defined by a probability distribution that depends on the parameters β and K, which represent, respectively, the inverse temperature and the interaction strength. Our main focus is on the asymptotic behavior of the magnetization m(β n ,K n ) for appropriate sequences (β n ,K n ) that converge to a second-order point or to the tricritical point of the model and that lie inside various subsets of the phase-coexistence region. The main result states that as (β n ,K n ) converges to one of these points (β,K), $m(\beta_{n},K_{n})\sim \bar{x}|\beta -\beta_{n}|^{\gamma}\rightarrow 0$ . In this formula γ is a positive constant, and $\bar{x}$ is the unique positive, global minimum point of a certain polynomial g. We call g the Ginzburg–Landau polynomial because of its close connection with the Ginzburg–Landau phenomenology of critical phenomena. For each sequence the structure of the set of global minimum points of the associated Ginzburg–Landau polynomial mirrors the structure of the set of global minimum points of the free-energy functional in the region through which (β n ,K n ) passes and thus reflects the phase-transition structure of the model in that region. This paper makes rigorous the predictions of the Ginzburg–Landau phenomenology of critical phenomena and the tricritical scaling theory for the mean-field Blume–Capel model.Publication Understanding temperature and chemical potential using computer simulations(2005-01-01) Tobochnik, J; Gould, H; Machta, JonathanSeveral Monte Carlo algorithms and applications that are useful for understanding the concepts of temperature and chemical potential are discussed. We then introduce a generalization of the demon algorithm that measures the chemical potential and is suitable for simulatingPublication High-precision measurement of the thermal exponent for the three-dimensional XY universality class(2006-01-01) Burovski, E; Machta, J; Prokof'ev, Nikolai; Svistunov, BorisSimulation results are reported for the critical point of the two-component ϕ4 field theory. The correlation-length exponent is measured to high precision with the result ν=0.6717(3). This value is in agreement with recent simulation results [Campostrini et al., Phys. Rev. B 63, 214503 (2001)] and marginally agrees with the most recent space-based measurements of the superfluid transition in 4He [Lipa et al., Phys. Rev. B 68, 174518 (2003)].Publication Numerical study of the three-dimensional random-field Ising model at zero and positive temperature(2006-01-01) Wu, Y; Machta, JonathanIn this paper the three-dimensional random-field Ising model is studied at both zero temperature and positive temperature. Critical exponents are extracted at zero temperature by finite size scaling analysis of large discontinuities in the bond energy. The heat capacity exponent α is found to be near zero. The ground states are determined for a range of external field and disorder strength near the zero temperature critical point and the scaling of ground state tilings of the field-disorder plane is discussed. At positive temperature the specific heat and the susceptibility are obtained using the Wang-Landau algorithm. It is found that sharp peaks are present in these physical quantities for some realizations of systems sized 163 and larger. These sharp peaks result from flipping large domains and correspond to large discontinuities in ground state bond energies. Finally, zero temperature and positive temperature spin configurations near the critical line are found to be highly correlated suggesting a strong version of the zero temperature fixed point hypothesis.