# "More is the Same; Phase Transitions and Mean Field Theories" by Leo P. Kadanoff

"More is the Same; Phase Transitions and Mean Field Theories" by Leo P. Kadanoff J Stat Phys (2009) 137: 777–797 DOI 10.1007/s10955-009-9814-1

В обзорной статье, на которую я давал ссылку пару дней назад, Лео Каданов утверждает, что фазовые переходы невозможны не только в одномерных системах (как написано у Ландау и Лифшица, https://duchifat.livejournal.com/1861048.html), но и в конечных системах.

1.5 More Is the Same; Infinitely More Is different

In discussing phase transitions, we must note a point that is fundamental to condensed matter physics. In the words of Anderson , more is different. The properties of systems containing infinitely many particles are qualitatively different from those of finite systems. In particular, phase transitions cannot occur in any finite system; they are solely a property of infinite systems.

To see this we follow Ehrenfest and Gibbs and define a phase transition as a point of singularity, i.e. sudden change. Let us see what this definition implies about any finite Ising model, one containing a finite number of spin variables. A phase transition occurs at points in the phase diagram where the free energy is a singular function of the thermodynamic variables within it. The partition function and the free energy are defined in (2). The former is the sum of the exponential of −H/T over all possible configurations. Such a sum of a finite number of exponentials is necessarily a positive quantity and one that is regular, i.e. not singular, for any finite value of K or h. Such a nonsingular quantity can have no sudden changes. Taking the logarithm of a positive quantity introduces no singularities, nor does division by a finite number, the temperature. Hence, we can conclude that, for the finite Ising model, the free energy is a non-singular function of K and h for all finite, real values of these parameters. By this argument the Ising model, as we have described it, can have no phase transitions. Further, there cannot be any phase transition in any finite system described by the Ising model or indeed any statistical system with everything in it being finite.
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Lev Landau  later argued that there was no phase transition in any finite-temperature system in one dimension. So, by that argument, the first situation in which we might expect to find a phase transition is in an infinite two-dimensional system.

5 This paper emphasizes that phase transitions in materials are connected with singularities created by the system’s infinite spacial extension. Other possible sources of singularities exist. On possibility, important in particle physics, is an ultraviolet divergence, that is an infinite number of degrees of freedom appearing within a finite volume. Another is an interaction which is infinitely strong. A third possibility is that the definition of the statistical average itself includes an infinity. This last possibility is realized in the microcanonical ensemble, which includes an infinitely sharp peak in energy .

Забавный у него стиль:

There is another way of knowing that the Ising model will have a phase transition. One can define a variety of dynamical models with the property that, if a finite system is run over a very long period of time, the model will reproduce the result of Gibbsian statistical mechanics. In running these models, if one is sufficiently patient the smaller systems will explore all the possible configurations and the average value of each of the spins will be zero. Thus, the finite system will not have a phase transition. As the size of the system gets larger, the system will tend to get stuck and explore only a limited subset of the possible configurations. The dependence of the configurations explored upon the size, shape, and couplings within the system is a major subject of contemporary, Twenty-First Century, exploration. I cannot fully review these dynamical investigations here. Nor can I discuss the analogous problems and results that arise in the experimental domain. I shall however give a tiny, superficial, overview.

In the case in which the system under study is large in only one of its directions and much smaller in its other dimensions, the system is said to be “one-dimensional”. In that case, as predicted by Landau, there tends to be a rather full examination of the phase space. In the case in which the system extends over a long distance in more than one of the possible directions, then the system can easily get stuck and explore only a limited portion of the available configuration space. However, “getting stuck” is not a simple easily understood event. There are at least three qualitatively different scenarios for a less than full exploration of the available configurations. In one case, independent of the starting state the entire Ising system will go into one of two possible “basins of attraction” defined by the two possible directions of the magnetization. This first scenario is the one described here, and closely follows the possible behavior of some real materials. In a second scenario, it is possible for the system to find itself in one of many different regions in “configuration space” and only explore that relatively small region, at least in any reasonable period of time. Over longer times, the size of the region will grow very slowly, but the region never encompasses most of the possible configurations. The repeated exploration of a slowly growing region of configurations is characteristic of a behavior described as “glassy”. Such glasses tend to occur in many materials with relatively strong interactions. They are believed to be in some cases a dynamical property of materials, and in others an equilibrium property described by an extension of Gibbsian statistical mechanics. Present-day condensed matter science does not understand glassy behavior. A third scenario has the system divide in a time-independent fashion into different regions, called domains or grains, each with its own phase.

Интуитивно мне кажется, что два параграфа выше - про эргодическую теорию (или даже про КАМ), но образования мне не хватает, а интуиции недостаточно, чтобы понять, что он конкретно имеет в виду.

О соотношении между стат. физикой и термодинамикой и о различии термина "редукция" в физике и в философии естествознания:

Philosophers of science will look at Maxwell’s application of thermodynamics with some interest. The philosophy literature contains a considerable discussion of Ernest Nagel’s principle of reduction  that describes how a more fundamental theory reduces a less fundamental one in an appropriate limit [22, 23]. This will occur when the ideas and laws of the reducing theory implies all the ideas and laws of the reduced theory .8 An example often employed is that thermodynamics might be a reduction of statistical mechanics, in the philosophers’ sense. However, here Maxwell extended a statistical calculation by using thermodynamics. I would worry about whether there is a simple process of reduction at work between statistical mechanics and thermodynamics, or perhaps between any two parts of science. One hint at possible complication in this case is the title of Erwin Schrödinger’s classic text “Statistical Thermodynamics” . Similar titles have been used by many other authors.

8 A physicist would use the word reduce differently, with the arrow going in the other direction. A physicist might say: Special relativity reduces to Galilean relativity when all speeds are small in comparison to the velocity of light. This difference in usage has been the source of some confusion.

Дальше у него про yp-e Ландау-Гинзбурга (без упоминания последнего по имени), про НИСТ с похвалой Сенгерс (непонятно, заслуженной или что-то из био самого Каданоффа) и, наконец, про Кеннета Вильсона (ренормализация). Что до Аннеке Сенгерс, я и не подозревал, что она такой крупный ученый (взаимодействовал с ней немного в НИСТе в 2007 году, мы пытались через нее заслать статью в ПНАС, но зарубил Израилашвили, которому она послала на рецензию). Что же они ей нобелевскую премию не дали? :)
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Согласно Карлу Юнгу, есть вертикальные причинные связи между событиями и есть горизонтальные - синхроничность. (Единственное, увлекаться поиском…

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Читаю вот этот странный сайт https://kniganews.org нигде не написано, кто автор. Явно человек хорошо понимаюший проблемы современной физики. Кто…

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Немного эзотерики в ленту. Если вы интересуетесь юнговской синхронией (это я тут с одной знакомой преподавательницей танцев из Питера обсуждал), то…