scholarly journals Analytical Expressions for Ising Models on High Dimensional Lattices

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1665
Author(s):  
Boris Kryzhanovsky ◽  
Leonid Litinskii ◽  
Vladislav Egorov
Keyword(s):  
Critical Point ◽  
Three Dimensional ◽  
Ising Models ◽  
High Dimensional ◽  
High Dimensions ◽  
Critical Temperatures ◽  
Dimensional Lattice ◽  
Long Range Interactions ◽  
Analytical Expressions ◽  
Good Agreement

We use an m-vicinity method to examine Ising models on hypercube lattices of high dimensions d ≥ 3. This method is applicable for both short-range and long-range interactions. We introduce a small parameter, which determines whether the method can be used when calculating the free energy. When we account for interaction with the nearest neighbors only, the value of this parameter depends on the dimension of the lattice d. We obtain an expression for the critical temperature in terms of the interaction constants that is in a good agreement with the results of computer simulations. For d = 5,6,7, our theoretical estimates match the numerical results both qualitatively and quantitatively. For d = 3,4, our method is sufficiently accurate for the calculation of the critical temperatures; however, it predicts a finite jump of the heat capacity at the critical point. In the case of the three-dimensional lattice (d = 3), this contradicts the commonly accepted ideas of the type of the singularity at the critical point. For the four-dimensional lattice (d = 4), the character of the singularity is under current discussion. For the dimensions d = 1,2 the m-vicinity method is not applicable.

scholarly journals MONTE-CARLO CALCULATIONS OF PHASE TRANSITION TEMPERATURE IN THE ISING MODEL WITH LONG RANGE INTERACTIONS

2017 ◽  
Vol 21 (6) ◽  
pp. 161-170
Author(s):  
A.A. Biryukov ◽  
Y.V. Degtyarova
Keyword(s):  
Phase Transition ◽  
Monte Carlo ◽  
Transition Temperature ◽  
Long Range ◽  
Phase Transition Temperature ◽  
Three Dimensional ◽  
Ising Models ◽  
Monte Carlo Calculations ◽  
Spin Interactions ◽  
Long Range Interactions

The article deals with two-dimensional and three-dimensional Ising models with the long-range spin interactions. The intensity of interaction between the spins relies decreasing with distance r as a power law r-d-σ with dimensional d and parameter σ. The research are conducted by Monte-Carlo method with Metropolis algorithm using parallel computing techniques. On the basis of nu- merical simulation the dependence of the phase transition temperature on the parameter σ is found. It is shown that at phase transition temperature decreases with increasing σ.

scholarly journals Exact Solution for Three-Dimensional Ising Model

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1837
Author(s):  
Degang Zhang
Keyword(s):  
Boundary Condition ◽  
Ising Model ◽  
Partition Function ◽  
Operator Algebras ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Critical Temperatures ◽  
Cubic Crystal ◽  
Infinite Crystal ◽  
Analytical Expressions

The three-dimensional Ising model in a zero external field is exactly solved by operator algebras, similar to the Onsager’s approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order–disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented.

Rapid Critical Point Drying of Tissues in a Parr Bomb

1972 ◽  
Vol 30 ◽  
pp. 400-401
Author(s):  
C.A. Baechler ◽  
W. C. Pitchford ◽  
J. M. Riddle ◽  
C.B. Boyd ◽  
H. Kanagawa ◽  
...  
Keyword(s):  
Critical Point ◽  
Critical Pressure ◽  
Soft Tissues ◽  
Biological Tissues ◽  
Critical Temperatures ◽  
Air Drying ◽  
The Past ◽  
Critical Point Drying ◽  
Great Stress ◽  
Infiltration Techniques

Preservation of the topographic ultrastructure of soft biological tissues for examination by scanning electron microscopy has been accomplished in the past by using lengthy epoxy infiltration techniques, or dehydration in ethanol or acetone followed by air drying. Since the former technique requires several days of preparation and the latter technique subjects the tissues to great stress during the phase change encountered during air-drying, an alternate rapid, economical, and reliable method of surface structure preservation was developed. Turnbill and Philpott had used a fluorocarbon for the critical point drying of soft tissues and indicated the advantages of working with fluids having both moderately low critical pressures as well as low critical temperatures. Freon-116 (duPont) which has a critical temperature of 19. 7 C and a critical pressure of 432 psi was used in this study.

Preface to the special issue, CMAM 2011, no. 3.

2011 ◽  
Vol 11 (3) ◽  
pp. 272
Author(s):  
Ivan Gavrilyuk ◽  
Boris Khoromskij ◽  
Eugene Tyrtyshnikov
Keyword(s):  
Separation Of Variables ◽  
Numerical Algorithms ◽  
Multilevel Methods ◽  
High Dimensional ◽  
Special Issue ◽  
High Dimensions ◽  
Guiding Principle ◽  
Tensor Methods ◽  
Closed World ◽  
The One

Abstract In the recent years, multidimensional numerical simulations with tensor-structured data formats have been recognized as the basic concept for breaking the "curse of dimensionality". Modern applications of tensor methods include the challenging high-dimensional problems of material sciences, bio-science, stochastic modeling, signal processing, machine learning, and data mining, financial mathematics, etc. The guiding principle of the tensor methods is an approximation of multivariate functions and operators with some separation of variables to keep the computational process in a low parametric tensor-structured manifold. Tensors structures had been wildly used as models of data and discussed in the contexts of differential geometry, mechanics, algebraic geometry, data analysis etc. before tensor methods recently have penetrated into numerical computations. On the one hand, the existing tensor representation formats remained to be of a limited use in many high-dimensional problems because of lack of sufficiently reliable and fast software. On the other hand, for moderate dimensional problems (e.g. in "ab-initio" quantum chemistry) as well as for selected model problems of very high dimensions, the application of traditional canonical and Tucker formats in combination with the ideas of multilevel methods has led to the new efficient algorithms. The recent progress in tensor numerical methods is achieved with new representation formats now known as "tensor-train representations" and "hierarchical Tucker representations". Note that the formats themselves could have been picked up earlier in the literature on the modeling of quantum systems. Until 2009 they lived in a closed world of those quantum theory publications and never trespassed the territory of numerical analysis. The tremendous progress during the very recent years shows the new tensor tools in various applications and in the development of these tools and study of their approximation and algebraic properties. This special issue treats tensors as a base for efficient numerical algorithms in various modern applications and with special emphases on the new representation formats.

A Free Energy Perturbation Approach to Estimate the Intrinsic Solubilities of Drug-like Small Molecules

2019 ◽  
Author(s):  
Sayan Mondal ◽  
Gary Tresadern ◽  
Jeremy Greenwood ◽  
Byungchan Kim ◽  
Joe Kaus ◽  
...  
Keyword(s):  
Solid State ◽  
Small Molecules ◽  
Three Dimensional ◽  
Aqueous Solubility ◽  
Computational Approach ◽  
Free Energy Perturbation ◽  
Perturbation Approach ◽  
Energy Perturbation ◽  
State Form ◽  
Good Agreement

<p>Optimizing the solubility of small molecules is important in a wide variety of contexts, including in drug discovery where the optimization of aqueous solubility is often crucial to achieve oral bioavailability. In such a context, solubility optimization cannot be successfully pursued by indiscriminate increases in polarity, which would likely reduce permeability and potency. Moreover, increasing polarity may not even improve solubility itself in many cases, if it stabilizes the solid-state form. Here we present a novel physics-based approach to predict the solubility of small molecules, that takes into account three-dimensional solid-state characteristics in addition to polarity. The calculated solubilities are in good agreement with experimental solubilities taken both from the literature as well as from several active pharmaceutical discovery projects. This computational approach enables strategies to optimize solubility by disrupting the three-dimensional solid-state packing of novel chemical matter, illustrated here for an active medicinal chemistry campaign.</p>

CALCULATION OF THE DEPENDENCE OF THE DISTANCE TO THE LOCATED OBJECT FROM THE CORNER POSITION OF THE VEHICLE LINE

2018 ◽  
pp. 14-18
Author(s):  
V. V. Artyushenko ◽  
A. V. Nikulin
Keyword(s):  
Real Time ◽  
Statistical Physics ◽  
Angular Position ◽  
Three Dimensional ◽  
Line Of Sight ◽  
Two Dimensional ◽  
Radar Station ◽  
Flight Mode ◽  
Vector Algebra ◽  
Analytical Expressions

To simulate echoes from the earth’s surface in the low flight mode, it is necessary to reproduce reliably the delayed reflected sounding signal of the radar in real time. For this, it is necessary to be able to calculate accurately and quickly the dependence of the distance to the object being measured from the angular position of the line of sight of the radar station. Obviously, the simplest expressions for calculating the range can be obtained for a segment or a plane. In the text of the article, analytical expressions for the calculation of range for two-dimensional and three-dimensional cases are obtained. Methods of statistical physics, vector algebra, and the theory of the radar of extended objects were used. Since the calculation of the dependence of the range of the object to the target from the angular position of the line of sight is carried out on the analytical expressions found in the paper, the result obtained is accurate, and due to the relative simplicity of the expressions obtained, the calculation does not require much time.

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