P. M. Chaikin, Princeton University, New Jersey, T. C. Lubensky, University of Pennsylvania. Publisher: Cambridge . pp i-vi. Access. PDF; Export citation. by: Chaikin, Paul M; Lubensky, T. C External-identifier: urn:acs6: principlesofcond00chai:pdf:ad0f63fc-1acb-4cde9-eb9edd The principal textbook is “Principle of Condensed Matter Physics” by Chaikin and . Lubensky, Cambridge University Press, This course has mainly the.
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PDF | On Jun 1, , Rudolf Podgornik and others published Principles of P. M. Chaikin and T. C. Lubensky, Cambridge University Press, Cambridge. Chaikin Lubensky - Principles of Condensed Matter portal7.info - Ebook download as PDF File .pdf) or read book online. P.M. Chaikin and T.C. Lubensky- Principles of Condensed Matter Physics - Free ebook download as PDF File .pdf) or read book online for free.
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In the last section the dynamics of complex magnetic metastability regions is studied numerically and the explanation of results is given.
The first sum in the r. Using the recursive structure of the Bethe lattice Fig. Let us now give some definitions and briefly discuss the problem of phase transitions on recursive lattices in terms of dynamical systems theory.
The thermodynamic properties of models defined on recursive lattices may be investigated by studying the dynamics of the corresponding recursive function. For the ferromagnetic Ising model, for example, this may be done as follows. In the absence of a magnetic field these two attracting fixed points correspond to two 4 possible ferromagnetic states with opposite magnetizations.
The other fixed point corresponds to a metastable state, which may be achieved by a sudden reversal in the sign of the magnetic field. The boundary of the metastability region may be found from the condition that one of the fixed points becomes neutral.
For more details about the dynamics of metastable states see the book by Chaikin and Lubensky . The critical temperature corresponds to the values of the magnetic field and temperature when the fixed points of recursive function f x are neutral and repelling.
The values of external parameters temperature, magnetic field, etc.
In the following section the Yang-Lee zeros of the 1D Q-state Potts model are investigated in terms of neutral fixed points. Analyzing Eq. It is noteworthy that using this equation one can study the Fisher and Potts zeros1 in the same way2. First of all note that Eq. This is the well known gap  in the distribution of Yang-Lee zeros.
This is in good agreement with recent studies by Glumac and Uzelac , and Kim and Creswick , where the Yang-Lee zeros of the 1D Potts model was studied by using the transfer matrix method.
Comparing our formula 13 with the corresponding formula 14 of Ref. It follows from the Eqs.
The same is true for the antiferromagnetic case. In this case the study of two maximal eigenvalues gives the same results as our method. The summary of results of the 1D ferromagnetic Potts model is given in Fig. The antiferromagnetic case may be studied in the same manner. The results are shown in Fig. The dashed areas in Fig.
Inside the metastability region there are two attracting fixed points and there is only one outside of it. The other fixed points are repelling.
Since there is no phase transition on the boundary of a metastability region it will not give rise to zeros of the partition function . The problem of finding the Yang-Lee zeros of models with first order phase transitions attracted much attention for many years. Recently, Biskup et al. Dolan et al. It is interesting to note that the locus of Fisher zeros on a Bethe lattice is identical to the corresponding random graph .
For our models the phase coexistence line is defined as a line in the complex plane, where the absolute values of the recursive function derivatives in two attracting fixed points are equal see also . Note that Eq. This was done for the first time by Bessis, Drouffe and Moussa  for the Ising model on Bethe lattice.
This is in a good agreement with the Yang-Lee theory . Inside it there are two attracting fixed points and others are repelling. It gives a numerical algorithm for searching neutral periodical points for the recursive functions like Eq. The algorithm is to find all critical points of the recursive function and investigate the convergence of all the orbits started at critical points critical orbits. If all critical orbits converge to any attracting periodical point one says that the recursive function has only attracting and repelling periodical points.
If at least one of the critical orbits does not converge, for example, after n iterations, one says that the recursive function has a neutral periodical point.
Of course, the last statement is not rigorous from the strong mathematical point of view since a weak convergence to an attracting periodical point is also possible. One can easily find all critical points of the mapping f from Eq.
It may be shown that these are the only critical points of the mapping f . In Fig. We have experimental evidence that in white regions all critical orbits converge.
This fact is known as the universality of the Mandelbrot set . One can see that Fig. By lowering Q the metastability regions become more and more complicated Figs. Note that the dynamics of the metastability regions remains the same if one fixes Q and changes the temperature. Note that in the one-dimensional case the condition of a phase transition is equivalent to the condition of existence of neutral fixed points.
We suppose that the box counting dimension of the Julia set of the recurrence relation should be a minimum for zeros of the partition function also. This may serve as a new criterion for studying zeros of the partition function for models on recursive lattices. In conclusion we observe that numerical methods proposed in this paper are generic and may be used for investigation of zeros of the partition function and metastability regions of other models on recursive lattices.
One of the authors R. Fauve and Prof. References  F. Wu, Rev. Potts, Proc. Aharony, P. Pfeuty, J. C: Solid State Phys. Lubensky, J.
Isaacson, Phys. Lee and C. Yang, Phys. Fisher, in Lectures in Theoretical Physics, edited by W.
Arndt, Phys. Monroe, J. Itzykson, R. Pearson, J. Zuber, Nucl. B [FS8], ; D. Ru- elle, Phys. Lu, F.
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