|
Ice-type Model
In statistical mechanics, the ice-type models or six-vertex models are a family of vertex models for crystal lattices with hydrogen bonds. The first such model was introduced by Linus Pauling in 1935 to account for the residual entropy of water ice. Variants have been proposed as models of certain ferroelectric and antiferroelectric crystals. In 1967, Elliott H. Lieb found the exact solution to a two-dimensional ice model known as "square ice". The exact solution in three dimensions is only known for a special "frozen" state. Description An ice-type model is a lattice model defined on a lattice of coordination number 4. That is, each vertex of the lattice is connected by an edge to four "nearest neighbours". A state of the model consists of an arrow on each edge of the lattice, such that the number of arrows pointing inwards at each vertex is 2. This restriction on the arrow configurations is known as the ice rule. In graph theoretic terms, the states are Eulerian orie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ..., information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation (graph Theory)
In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph. Oriented graphs A directed graph is called an oriented graph if none of its pairs of vertices is linked by two mutually symmetric edges. Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of and may be arrows of the graph). A tournament is an orientation of a complete graph. A polytree is an orientation of an undirected tree. Sumner's conjecture states that every tournament with vertices contains every polytree with vertices. The number of non-isomorphic oriented graphs with vertices (for ) is : 1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, … . Tournaments are in one-to-one correspondence with complete directed graphs (graphs in which there is a directed edge in one or both directions between every pair of distinct vertices). A complete directed graph can be con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William F
William is a masculine given name of Germanic languages, Germanic origin. It became popular in England after the Norman Conquest, Norman conquest in 1066,All Things William"Meaning & Origin of the Name"/ref> and remained so throughout the Middle Ages and into the modern era. It is sometimes abbreviated "Wm." Shortened familiar versions in English include Will (given name), Will or Wil, Wills, Willy, Willie, Bill (given name), Bill, Billie (given name), Billie, and Billy (name), Billy. A common Irish people, Irish form is Liam. Scottish people, Scottish diminutives include Wull, Willie or Wullie (as in Oor Wullie). Female forms include Willa, Willemina, Wilma (given name), Wilma and Wilhelmina (given name), Wilhelmina. Etymology William is related to the German language, German given name ''Wilhelm''. Both ultimately descend from Proto-Germanic ''*Wiljahelmaz'', with a direct cognate also in the Old Norse name ''Vilhjalmr'' and a West Germanic borrowing into Medieval Latin ''Wil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Moment
In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude of torque the object experiences in a given magnetic field. When the same magnetic field is applied, objects with larger magnetic moments experience larger torques. The strength (and direction) of this torque depends not only on the magnitude of the magnetic moment but also on its orientation relative to the direction of the magnetic field. Its direction points from the south pole to the north pole of the magnet (i.e., inside the magnet). The magnetic moment also expresses the magnetic force effect of a magnet. The magnetic field of a magnetic dipole is proportional to its magnetic dipole moment. The dipole component of an object's magnetic field is symmetric about the direction of its magnetic dipole moment, and decreases as the inverse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrical Frustration
In condensed matter physics, geometrical frustration (or in short, frustration) is a phenomenon where the combination of conflicting inter-atomic forces leads to complex structures. Frustration can imply a plenitude of distinct ground states at absolute zero, zero temperature, and usual thermal ordering may be suppressed at higher temperatures. Much-studied examples include amorphous materials, glasses, and dilute magnets. The term ''frustration'', in the context of magnetism, magnetic systems, was introduced by Gerard Toulouse in 1977. Frustrated magnetism, magnetic systems had been studied even before. Early work includes a study of the Ising model on a triangular lattice with nearest-neighbor Spin (physics), spins coupled Antiferromagnetism, antiferromagnetically, by Gregory Wannier, G. H. Wannier, published in 1950. Related features occur in magnets with ''competing interactions'', where both ferromagnetic as well as antiferromagnetic couplings between pairs of Spin (physics), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Ice
A spin ice is a magnetic substance that does not have a single minimal-energy state. It has magnetic moments (i.e. "spin") as elementary degrees of freedom which are subject to frustrated interactions. By their nature, these interactions prevent the moments from exhibiting a periodic pattern in their orientation down to a temperature much below the energy scale set by the said interactions. Spin ices show low-temperature properties, residual entropy in particular, closely related to those of common crystalline water ice. The most prominent compounds with such properties are dysprosium titanate (Dy2Ti2O7) and holmium titanate (Ho2Ti2O7). The orientation of the magnetic moments in spin ice resembles the positional organization of hydrogen atoms (more accurately, ionized hydrogen, or protons) in conventional water ice (see figure 1). Experiments have found evidence for the existence of deconfined magnetic monopoles in these materials, with properties resembling those of the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermodynamic Limit
In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the Limit (mathematics), limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundell and K.M. Blundell, "Concepts in Thermal Physics", Oxford University Press (2009) The thermodynamic limit is defined as the limit of a system with a large volume, with the particle density held fixed: : N \to \infty,\, V \to \infty,\, \frac N V =\text In this limit, macroscopic thermodynamics is valid. There, thermal fluctuations in global quantities are negligible, and all List of thermodynamic properties, thermodynamic quantities, such as pressure and energy, are simply functions of the thermodynamic variables, such as temperature and density. For example, for a large volume of gas, the fluctuations of the total internal energy are negligible and can be ignored, and the average internal energy can be predicted fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the molar gas constant, in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating Johnson–Nyquist noise, thermal noise in resistors. The Boltzmann constant has Dimensional analysis, dimensions of energy divided by temperature, the same as entropy and heat capacity. It is named after the Austrian scientist Ludwig Boltzmann. As part of the 2019 revision of the SI, the Boltzmann constant is one of the seven "Physical constant, defining constants" that have been defined so as to have exact finite decimal values in SI units. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly joules per kelvin, with the effect of defining the SI unit ke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Function (statistical Mechanics)
In physics, a partition function describes the statistics, statistical properties of a system in thermodynamic equilibrium. Partition functions are function (mathematics), functions of the thermodynamic state function, state variables, such as the temperature and volume. Most of the aggregate thermodynamics, thermodynamic variables of the system, such as the energy, total energy, Thermodynamic free energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless. Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular Thermodynamic free energy, free energy). The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the Environment (systems), environment at fixed temperature, volume, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ice Ih
Variations in pressure and temperature give rise to different phases of ice, which have varying properties and molecular geometries. Currently, twenty-one phases, including both crystalline and Amorphous solid, amorphous ices have been observed. In modern history, phases have been discovered through scientific research with various techniques including pressurization, force application, nucleation agents, and others. On Earth, most ice is found in the hexagonal Ice Ih phase. Less common phases may be found in the atmosphere and underground due to more extreme pressures and temperatures. Some phases are manufactured by humans for nano scale uses due to their properties. In space, amorphous ice is the most common form as confirmed by observation. Thus, it is theorized to be the most common phase in the universe. Various other phases could be found naturally in astronomical objects. Theory Most liquids under increased pressure freeze at ''higher'' temperatures because the pressu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
