Ising model
The Ising model is of Ernst Ising, at the suggestion of his doctor father Wilhelm Lenz in 1924 for the first time more closely -studied model in theoretical physics. It describes in particular the ferromagnetism in solids ( crystals ). The Ising model is one of the most studied models of statistical physics.
Definition
In the model it is assumed that the spins that determine the magnetic moment of atoms or ions can assume only two discrete states (spin value ). The direction in space remains open; So are vectors ( to stay in the classic picture, or quantum mechanically to vector operators ).
The general expression of energy (or Hamiltonian ) for such a situation is given by the Heisenberg model:
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Referred to this
- A ( multi-component ) spin of the atom in place of the crystal lattice,
- The magnetic field,
- The Hamiltonian and
- The coupling constant ( exchange coupling ) between the spins at the places and. It indicates the interaction strength.
The point marks the so-called scalar product.
Organising the model, however, the number of components of the spinning is reduced to one ( i.e. parallel or anti- parallel to an easy axis - in this case the z- axis).
:
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Often is also assumed that only non-zero neighboring spins. If the exchange coupling is positive, then one speaks of a ferromagnetic coupling; if it is negative, it is called antiferromagnetic. In ferromagnets and antiferromagnets dominates each sign; both signs are used in the so-called spin glasses the same common.
By suitable choice of the interactions already mentioned spin glasses can, inter alia, (this is a random variable ), dilute magnets with interesting critical properties or spatially modulated magnetic structures ( in this case lie before competing couplings, see annni model) are modeled. In general, the model describes the Organising magnetic systems at low temperatures, however, are broken by thermal fluctuations at higher temperatures, a phase transition takes place. A comprehensive theoretical analysis of phase transitions provides the renormalization group, 1982 received the Nobel Prize in Physics for the Kenneth G. Wilson.
In the one-dimensional Ising chain with sufficiently short-range interactions, however, observed no phase transition. This had Ernst Ising must determine in his doctoral thesis with regret. The exact solution of the two-dimensional Ising model with nearest neighbor interaction and vanishing magnetic field was first calculated in 1944 by Lars Onsager, but not published due to disinterest - Onsager was known only to published works that were really important to him. The publication came only in 1952, after it was Chen Ning Yang managed to reproduce Onsagers solution. It turned out that in this case, a phase transition occurs.
For the three-dimensional Ising model with interactions between neighboring spins, there is no analytical exact solution. Its properties may be defined using the molecular field approximation (or Landau theory ), Monte Carlo simulations, series expansions or other numerical solution methods compute.
The Ising model is considered because of its conceptual simplicity and its many properties as " Drosophila " of statistical physics. It has also found applications in many areas of science, to biology and brain research. The almost programmatic statement of Michael E. Fisher, Ising models quietly thrive ' (about: Ising models are still growing ') will probably remain valid for many years.
A generalization of the Ising model provides the Potts model or the Markov network.
Simplified representation
This section discusses the simplest Ising model: no external magnetic field, interaction only between nearest neighbors (left, right, up, down ). Opposite neighbor spins provide an energy contribution, parallel spins do not contribute.
The aim is to present the basic foundations for a wider readership.
Energy, heat, probability
The picture shows symbolically a tiny " magnets " of 25 " iron atoms ". Iron atoms behave like tiny magnets. The magnetic field of the magnet is the sum total of the magnetic fields that emanate from the individual atoms, the fields opposite cancel aligned atoms to each other.
Five of the atoms ( black ) are aligned in a direction here, the remaining 20 ( white) in the other direction. The net magnetization is thus units. A specific black and white pattern is referred to as the state of the magnet.
The 14 red edges show oppositely oriented neighbors. Each red edge corresponds to an amount of energy stored in the magnet, which is called (this is not for the energy unit Joule, but simply a characteristic of the particular material ).
Each red edge reduces the probability of finding the state of nature, and the more so the colder it is. You calculate this by dividing the probability of the state " all atoms in the same direction " for each red edge once with multiplied. It is the product of the temperature in Kelvin and the Boltzmann constant.
Example: On a warm summer day ( 27 degrees Celsius ) results in a material whose value is 0.0595 electron volts, each red edge a probability reduction by a factor of 10 on cooling to minus 123 degrees Celsius, the factor is already at minus 100 and 173 degrees even 1000th
What has been said relates to the probability of an individual state. It is usually very small. Now, there is usually also a very large number of states, which establish a certain magnetization strength of the magnet ( number of black squares) (think of the many ways to fill out a lottery ticket ).
The large number of conditions can compensate for the small probability of each state. Indeed, there are usually at a given temperature, a certain intensity of magnetisation, which significantly outperforms all others in probability. This magnetization is almost exclusively encountered. With increasing temperature, it shifts from " fully magnetized " " demagnetized " to.
Extreme temperatures
In order to find a sense of the importance of the foregoing, first consider the limiting cases of very high and very low temperature. Contrary to intuition, the calculations are thereby impeded not by large numbers, but even so simple that you already comes through " mental calculation " results.
At extremely low temperatures ( temperature, absolute zero approaches ) the probability factor is so small that no state can be encountered except " all black" or " all white " ever. The magnet thus takes on its full magnetization.
At extremely high temperatures, however, the probability factor of the number 1 is always similar, so that it does not lead to reduction in probability and all states are equally likely. Then, for each magnetisation, the sheer number of them realized states, and that is just for " 50 % white - 50 % black " at the highest. The magnet is effectively demagnetized.
Moderate temperature
The pictured condition with a different atom has four red edges. At a value of 0.0017 eV this is a condition ten times less likely than the full magnetization ( at 27 degrees Celsius). However, there are 25 ways to leave exactly one atom differ, and so a magnetization of 23 units ( 24-1 opposite ) 2.5 times as likely as the full magnetization.
Critical temperature
The collapse of magnetism occurs already at a finite temperature, the so-called critical temperature to. To justify this requires extensive mathematical analyzes which can not be performed here.
Occur near the critical temperature, however, "interesting" pattern (with respect to black -and-white distribution).
Structure formation
On the way from absolute zero to infinite temperature you can go from perfect order to perfect noise.
In between, one finds "interesting" pattern. For a qualitative justification is given.
With respect to the magnetization value is a compromise is emerging between low state probability and great condition number. Analogously, when the temperature and the magnetization are given, with respect to the scattering argue black and white squares.
Although an arbitrarily picked as compact structure has fewer red edges ( and is therefore more likely) than an arbitrarily singled out loosened structure; but because there is more loosened structures, the property " loosened " its total probable. Thus we will find a compromise that is neither very compact yet completely torn, just an "interesting " structure.
Applications and interpretations
So famous is the " magnetic" interpretation of the Ising model, the spin values show "up" or "down". But also for other binary problems, you might the Ising model: A prominent example is the so-called " Ising lattice gas ", which can be used for modeling of liquids: One case considers a grid whose squares either "occupied" or " unoccupied " can be, depending on whether the lattice site of the associated Ising spin the value 1 or -1. It is also clear that one can describe the so-called spin glasses with the Ising model, namely the energy, the s- variables denote the Ising spins and accept the Ji, k fixed, but random values . Very little is known, however, that this Hamiltonian is also an elementary interpretation has, which delivers a highly simplified model of quantum chromodynamics: One can namely the s- variables as "quarks" and the Ji, k as " gluons " interpret if you fluctuate both sizes leaves. However, one must still add those referred to as Wilson loop variables gluon -gluon couplings of the form in this case the Hamiltonian. One then obtains " gauge-invariant models," which uncorrelated binary variables and the coupled so-called gauge transformations, suffice; that is, the Hamiltonian remains invariant in these transformations, such as the Lagrangian of quantum chromodynamics over transformations with elements of the group SU (3) invariant remains, which are here replaced by the ε - variables. With this model - a kind of " Ising Lattice QCD " - the later so-called lattice gauge theory was introduced. The relevant publication to comes from Franz Wegner.
Another possible application is the simulation of phase transitions by nucleation. Homogeneous nucleation corresponds almost exactly to the ferromagnetism in modeling - for heterogeneous nucleation some small changes need to be made.
The first sum is in this case again, the interaction between neighbors - the second summation of " i, j ", however, is to interact with a boundary surface. It turns out that in the area of such boundary surfaces, a core critical size is many times created faster - based on were also simulations for nucleation performed on a porous surface, the result of which was that a certain size of the pores must be given to ensure the fastest possible nucleation (usually this is the case most likely to be irregular pores). Large pores in the portion of the boundary surfaces is small - there is no longer characterized Nukleationskern critical size in the pore - where the pore is small, however, the initiation of a phase transition from the upper edge is less likely to be off.