a rigorous theory of finite size scaling at first order phase transitions a large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered the dependence of the finite volume magnetizationm per h l in cubes of sizel d under periodic boundary conditions on the external fieldh is analyzed for the case where two phases coexist at the infinite volume transition pointh t we prove that independent of the details of the model the finite volume magnetization per lattice site behaves likem per h t m tanh ml d h ht withm per h denoting the infinite volume magnetization and m 1 2 m per h t 0 m per h t 0 introducing the finite size transition pointh m l as the point where the finite volume susceptibility attains the maximum we show that in the case of asymmetric field driven transitions its shift ish t h m l o l 2d in contrast to claims in the literature starting from the obvious observation that the number of stable phases has a local maximum at the transition point we propose a new way of determining the pointh t from finite size data with a shift that is exponentially small inl finally the finite size effects are discussed also in the case where more than two phases coexist

finite size scaling for potts models recently borgs and koteck developed a rigorous theory of finite size effects near first order phase transitions here we apply this theory to the ferromagneticq state potts model which forq large andd 2 undergoes a first order phase transition as the inverse temperature is varied we prove a formula for the internal energy in a periodic cube of side lengthl which describes the rounding of the infinite volume jump e in terms of a hyperbolic tangent and show that the position of the maximum of the specific heat is shifted by m l inq e l d o l 2d with respect to the infinite volume transition point t we also propose an alternative definition of the finite volume transition temperature t l which might be useful for numerical calculations because it differs only by exponentially small corrections from t

baxter wu model with fixed magnetization abstract we study baxter wu triangular model with fixed magnetization in the framework of canonical and microcanonical ensemble constructing the density of states by wang landau algorithm we use an approximation similar to a recently developed scheme critical minimum energy subspace in this scheme the sampling is not extended to the whole energy spectrum for the given magnetization but it is restricted in a properly determined part of it we have studied lattices with linear size up to l 840 our conclusions corroborate analogous conclusions for the square ising model an unconventional temperature driven first order phase transition take place for lattice sizes above some threshold but this transition ceases to exist at the thermodynamic limit

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

three dimensional antiferromagnetic q state potts models application of the wang landau algorithm we apply a newly proposed monte carlo method the wang landau algorithm to the study of the three dimensional antiferromagnetic q state potts models on a simple cubic lattice we systematically study the phase transition of the models with q 3 4 5 and 6 we obtain the finite temperature phase transition for q 3 and 4 whereas the transition temperature is down to zero for q 5 for q 6 there exists no order for any temperature we also study the ground state properties the size dependence of the ground state entropy is investigated we find that the ground state entropy is larger than the contribution from the typical configurations of the broken sublattice symmetry state for q 3 the same situations are found for q 4 5 and 6

determining the density of states for classical statistical models a random walk algorithm to produce a flat histogram we describe an efficient monte carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models the density of states is modified at each step when the energy level is visited to produce a flat histogram by carefully controlling the modification factor we allow the density of states to converge to the true value very quickly even for large systems from the density of states at the end of the random walk we can estimate thermodynamic quantities such as internal energy and specific heat capacity by calculating canonical averages at any temperature using this method we not only can avoid repeating simulations at multiple temperatures but we can also estimate the free energy and entropy quantities that are not directly accessible by conventional monte carlo simulations this algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability as with the multicanonical monte carlo technique our method overcomes the tunneling barrier between coexisting phases at first order phase transitions in this paper we apply our algorithm to both first and second order phase transitions to demonstrate its efficiency and accuracy we obtained direct simulational estimates for the density of states for two dimensional ten state potts models on lattices up to 200 x 200 and ising models on lattices up to 256 x 256 our simulational results are compared to both exact solutions and existing numerical data obtained using other methods applying this approach to a three dimensional j spin glass model we estimate the internal energy and entropy at zero temperature and using a two dimensional random walk in energy and order parameter space we obtain the rough canonical distribution and energy landscape in order parameter space preliminary data suggest that the glass transition temperature is about 1 2 and that better estimates can be obtained with more extensive application of the method this simulational method is not restricted to energy space and can be used to calculate the density of states for any parameter by a random walk in the corresponding space

understanding and improving the wang landau algorithm we present a mathematical analysis of the wang landau algorithm prove its convergence and identify sources of errors and strategies for optimization in particular we found the histogram increases uniformly with small fluctuations after a stage of initial accumulation and the statistical error is found to scale as sqrt mathrm ln phantom rule 0 2em 0ex f with the modification factor f this has implications for strategies for obtaining fast convergence

continuous phase transitions with a convex dip in the microcanonical entropy the appearance of a convex dip in the microcanonical entropy of finite systems usually signals a first order transition however a convex dip also shows up in some systems with a continuous transition as for example in the baxter wu model and in the four state potts model in two dimensions we demonstrate that the appearance of a convex dip in those cases can be traced back to a finite size effect the properties of the dip are markedly different from those associated with a first order transition and can be understood within a microcanonical finite size scaling theory for continuous phase transitions results obtained from numerical simulations corroborate the predictions of the scaling theory

efficient multiple range random walk algorithm to calculate the density of states we present a new monte carlo algorithm that produces results of high accuracy with reduced simulational effort independent random walks are performed concurrently or serially in different restricted ranges of energy and the resultant density of states is modified continuously to produce locally flat histograms this method permits us to directly access the free energy and entropy is independent of temperature and is efficient for the study of both 1st order and 2nd order phase transitions it should also be useful for the study of complex systems with a rough energy landscape

lack of self averaging of the specific heat in the three dimensional random field ising model we apply the recently developed critical minimum energy subspace scheme for the investigation of the random field ising model we point out that this method is well suited for the study of this model the density of states is obtained via the wang landau and broad histogram methods in a unified implementation by employing the n fold version of the wang landau scheme the random fields are obtained from a bimodal distribution h i ifmmode pm else textpm fi 2 and the scaling of the specific heat maxima is studied on cubic lattices with sizes ranging from l 4 to l 32 observing the finite size scaling behavior of the maxima of the specific heats we examine the question of saturation of the specific heat the lack of self averaging of this quantity is fully illustrated and it is shown that this property may be related to the question mentioned above

entropic sampling of simple polymer models within wang landau algorithm in this paper we apply a new simulation technique proposed in wang and landau wl 2001 phys rev lett 86 2050 to sampling of three dimensional lattice and continuous models of polymer chains distributions obtained by homogeneous unconditional random walks are compared with results of entropic sampling es within the wl algorithm while homogeneous sampling gives reliable results typically in the range of 4 5 orders of magnitude the wl entropic sampling yields them in the range of 20 30 orders and even larger with comparable computer effort a combination of homogeneous and wl sampling provides reliable data for events with probabilities down to 10 35 for the lattice model we consider both the athermal case self avoiding walks saws and the thermal case when an energy is attributed to each contact between nonbonded monomers in a self avoiding walk for short chains the simulation results are checked by comparison with the exact data in wl calculations for chain lengths up to n 300 scaling relations for saws are well reproduced in the thermal case distribution over the number of contacts is obtained in the n range up to n 100 and the canonical averages internal energy heat capacity excess canonical entropy mean square end to end distance are calculated as a result in a wide temperature range the continuous model is studied in the athermal case by sorting conformations of a continuous phantom freely joined n bonded chain with a unit bond length over a stochastic variable the minimum distance between nonbonded beads we determine the probability distribution for the n bonded chain with hard sphere monomer units over its diameter a in the complete diameter range 0 a 2 within a single es run this distribution provides us with excess specific entropy for a set of diameters a in this range calculations were made for chain lengths up to n 100 and results were extrapolated to n for a in the range 0 a 1 25

finite size scaling analysis of the critical behavior of the baxter wu model abstract we use the recently developed critical minimum energy subspace crmes approximation scheme to study the critical behavior of the baxter wu model this scheme uses only a properly determined part of the energy spectrum and allows us to obtain high accuracy for relatively large systems with considerably reduced computational effort the density of states is constructed by a multi range wang landau sampling and from this we obtain the critical properties of specific heat and of the reduced binder cummulant the good agreement of our results with existing exact solutions demonstrates the accuracy of our approximation technique

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

avoiding boundary effects in wang landau sampling a simple modification of the wang landau sampling algorithm removes the systematic error that occurs at the boundary of the range of energy over which the random walk takes place in the original algorithm

new monte carlo algorithm entropic sampling we present a new monte carlo sampling algorithm with which one can obtain any desired distribution of the sampling in one monte carlo simulation the free energy and the entropy of a system can thus be obtained from a simple exercise of this algorithm the main idea is to sample directly the entropy of a system at infinite temperature importance sampling is shown to be a particular case of the new algorithm the algorithm is tested against the exact partition function of the ital l 4 simple cubic ising model a comparison with the multicanonical ensemble for the ital l 12 ital q 10 potts model shows that the new algorithm is more general and more efficient

generalization of the wang landau method for off lattice simulations we present a rigorous derivation for off lattice implementations of the so called random walk algorithm recently introduced by wang and landau phys rev lett 86 2050 2001 originally developed for discrete systems the algorithm samples configurations according to their inverse density of states using monte carlo moves the estimate for the density of states is refined at each simulation step and is ultimately used to calculate thermodynamic properties we present an implementation for atomic systems based on a rigorous separation of kinetic and configurational contributions to the density of states by constructing a uniform ensemble for configurational degrees of freedom in which all potential energies volumes and numbers of particles are equally probable we establish a framework for the correct implementation of simulation acceptance criteria and calculation of thermodynamic averages in the continuum case to demonstrate the generality of our approach we perform sample calculations for the lennard jones fluid using two implementation variants and in both cases find good agreement with established literature values for the vapor liquid coexistence locus

estimation of critical behavior from the density of states in classical statistical models we present a simple and efficient approximation scheme which greatly facilitates the extension of wang landau sampling or similar techniques in large systems for the estimation of critical behavior the method presented in an algorithmic approach is based on a very simple idea familiar in statistical mechanics from the notion of thermodynamic equivalence of ensembles and the central limit theorem it is illustrated that we can predict with high accuracy the critical part of the energy space and by using this restricted part we can extend our simulations to larger systems and improve the accuracy of critical parameters it is proposed that the extensions of the finite size critical part of the energy space determining the specific heat satisfy a scaling law involving the thermal critical exponent the method is applied successfully for the estimation of the scaling behavior of specific heat of both square and simple cubic ising lattices the proposed scaling law is verified by estimating the thermal critical exponent from the finite size behavior of the critical part of the energy space the density of states of the zero field ising model on these lattices is obtained via a multirange wang landau sampling

a viscosity approach to infinite dimensional hamilton jacobi equations arising in optimal control with state constraints we consider nonlinear optimal control problems with state constraints and nonnegative cost in infinite dimensions where the constraint is a closed set possibly with empty interior for a class of systems with a maximal monotone operator and satisfying certain stability properties of the set of trajectories that allow the value function to be lower semicontinuous we prove that the value function is a viscosity solution of the bellman equation and is in fact the minimal nonnegative supersolution

entropic sampling via wang landau random walks in dominant energy subspaces dominant energy subspaces of statistical systems are defined with the help of restrictive conditions on various characteristics of the energy distribution such as the probability density and the fourth order binder s cumulant our analysis generalizes the ideas of the critical minimum energy subspace crmes technique applied previously to study the specific heat s finite size scaling here we illustrate alternatives that are useful for the analysis of further finite size anomalies and the behavior of the corresponding dominant subspaces is presented for the two dimensional 2d baxter wu and the 2d and 3d ising models in order to show that a crmes technique is adequate for the study of magnetic anomalies we study and test simple methods which provide the means for an accurate determination of the energy order parameter e m histograms via wang landau random walks the 2d ising model is used as a test case and it is shown that high level wang landau sampling schemes yield excellent estimates for all magnetic properties our estimates compare very well with those of the traditional metropolis method the relevant dominant energy subspaces and dominant magnetization subspaces scale as expected with exponents alpha nu and gamma nu respectively using the metropolis method we examine the time evolution of the corresponding dominant magnetization subspaces and we uncover the reasons behind the inadequacy of the metropolis method to produce a reliable estimation scheme for the tail regime of the order parameter distribution

finite size scaling and surface tension from effective one dimensional systems we develop a method for precise asymptotic analysis of partition functions near first order phase transitions working in a 1 dimensional cylinder of volumel l t we show that leading exponentials int can be determined from a simple matrix calculation providedt v logl through a careful surface analysis we relate the off diagonal matrix elements of this matrix to the surface tension andl while the diagonal matrix elements of this matrix are related to the metastable free energies of the model for the off diagonal matrix elements which are related to the crossover length from hypercubic l t to cylindrical t scaling this includes a determination of the pre exponential power ofl as a function of dimension the results are applied to supersymmetric field theory and in a forthcoming paper to the finite size scaling of the magnetization and inner energy at field and temperature driven first order transitions in the crossover region from hypercubic to cylindrical scaling

monte carlo study of the pure and dilute baxter wu model we studied the pure and dilute baxter wu bw models using the wang landau wl sampling method to calculate the density of states dos we first used the exact result for the dos of the ising model to test our code then we calculated the dos of the dilute ising model to obtain a phase diagram in good agreement with previous studies we calculated the energy distribution together with its first second and fourth moments to give the specific heat and the energy fourth order cumulant better known as the binder parameter for the pure bw model for small samples the energy distribution displayed a doubly peaked shape and finite size scaling analysis showed the expected reciprocal scaling of the positions of the peaks with l the energy distribution yielded the expected bw 2 3 critical exponent for the specific heat the binder parameter minimum appeared to scale with lattice size l with an exponent b equal to the specific heat exponent its location temperature showed a large correction to scaling term 1 0 248 0 025 for the dilute bw model we found a clear crossover to a single peak in the energy distribution even for small sizes and the expected 0 was recovered

on the new barrier function and specialized algorithms for a class of semidefinite programs semidefinite programs sdps arising from the kalman yakubovich popov kyp lemma are frequently encountered in systems robustness analysis filter design and other control signal processing related applications these programs possess a special structure that can be exploited to construct specialized algorithms that substantially outperform general purpose sdp solvers in this paper a new interior path following algorithm that utilizes this structure is proposed the main idea behind the algorithm is a new barrier function for these specially structured sdps convergence of the new algorithm is shown and a measure of the accuracy of suboptimal solutions produced by the algorithm is provided the algorithm is tested in numerical experiments and the results indicate that the new algorithm is indeed favorable against general purpose sdp solvers in many circumstances

first order transition features of the triangular ising model with nearest and next nearest neighbor antiferromagnetic interactions abstract we implement a new and accurate numerical entropic scheme to investigate the first order transition features of the triangular ising model with nearest neighbor j nn and next nearest neighbor j nnn antiferromagnetic interactions in ratio r j nn j nnn 1 important aspects of the existing theories of first order transitions are briefly reviewed tested on this model and compared with previous work on the potts model using lattices with linear sizes l 30 40 100 120 140 160 200 240 360 and 480 we estimate the thermal characteristics of the present weak first order transition our results improve the original estimates of rastelli et al and verify all the generally accepted predictions of the finite size scaling theory of first order transitions including transition point shifts thermal and magnetic anomalies however two of our findings are not compatible with current phenomenological expectations the behavior of transition points derived from the number of phases parameter is not in accordance with the theoretically conjectured exponentially small shift behavior and the well known double gaussian approximation does not correctly describe higher correction terms of the energy cumulants it is argued that this discrepancy has its origin in the commonly neglected contributions from domain wall corrections

finite size scaling for first order phase transitions rigorous results abstract recent rigorous results on finite size effects for lattice models are reviewed we consider symmetric and asymmetric first order phase transitions in cubic and cylindrical geometries and discuss the determination of transition points from finite size data

a unified approach to phase diagrams in field theory and statistical mechanics we construct the phase diagram of any system which admits a low temperature polymer or cluster expansion such an expansion turns the system into a hard core interacting contour model with small but not necessarily positive activities the method uses some of zahradnik s ideas z1 but applies equally well to systems with complex interactions we give two applications first to low temperaturep 2 models with complex couplings and second to a computation of asymptotics of partition functions in periodic volumes if the index of a supersymmetric field theory is known the second application would help determine the number of phases in infinite volume