General form of the probability distribution function (PDF) of the rescaled fractal dimension characterizing spatial structures of local order parameters in finite but large systems near the critical point is derived. To obtain the PDF, we assume that (i) sample-to-sample fluctuations of global order parameters are described by the generalized Gumbel distribution function, which is known to be valid for many of long-range correlated systems such as critical systems and (ii) spatial structures of local order parameters are always fractal at the critical point. From the obtained PDF, it is elucidated that fluctuations of fractal dimensions do not exhibit a universal character in contrast to the case of fluctuating global order parameters. In addition, we perform numerical calculations of the PDF for two-dimensional percolation systems near the critical concentration and confirm that the analytically calculated PDF well describes numerical results. We also discuss the influence of non-fractal critical clusters neglected under the assumption (ii).General form of the probability distribution function (PDF) of the rescaled fractal dimension characterizing spatial structures of local order parameters in finite but large systems near the critical point is derived. To obtain the PDF, we assume that (i) sample-to-sample fluctuations of global order parameters are described by the generalized Gumbel distribution function, which is known to be valid for many of long-range correlated systems such as critical systems and (ii) spatial structures of local order parameters are always fractal at the critical point. From the obtained PDF, it is elucidated that fluctuations of fractal dimensions do not exhibit a universal character in contrast to the case of fluctuating global order parameters. In addition, we perform numerical calculations of the PDF for two-dimensional percolation systems near the critical concentration and confirm that the analytically calculated PDF well describes numerical results. We also discuss the influence of non-fractal critical clusters neglected under the assumption (ii).

Journal of the Physical Society of JapanJul 10, 20093 citations

Correlation Function near the Critical Point

P. Gluck

physica status solidi (b)Aug 1, 19730 citations

Experimental investigation of the correlation function near the critical point

Mikhail A. Anisimov, A. M. Evtyushenkov, Yu. F. Kiyachenko +1 more

Sep 1, 19741 citation

Thermodynamic Structure of Correlation Functions near a Critical Point in Multicomponent Systems

Luciano Mistura

A simple generalization of the Griffiths and Wheeler postulates for the thermodynamic behavior near a critical point is introduced to describe fluctuations in local dynamical variables. It is proposed that in the classical Hilbert space of the dynamical variables there are only two linearly independent vectors, related to the shape of the coexistence surface and the critical surface, which contain critical fluctuations. It is shown that, as a consequence of the affine structure of the field space, the above assumption leads to the following asymptotic property for the correlation function ${f}_{\mathrm{ab}}(q, \ensuremath{\omega})$ of any two local operators $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{a}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{b}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$: ${f}_{\mathrm{ab}}(q, \ensuremath{\omega})={a}_{1}(q, \ensuremath{\omega}){b}_{1}(q, \ensuremath{\omega}){f}_{11}(q, \ensuremath{\omega})+{a}_{2}(q, \ensuremath{\omega}){b}_{2}(q, \ensuremath{\omega}){f}_{22}(q, \ensuremath{\omega})+{a}_{3}(q, \ensuremath{\omega}){b}_{3}(q, \ensuremath{\omega}){f}_{33}(q, \ensuremath{\omega})+\dots{}$Here $q$ is the wave number, $\ensuremath{\omega}$ is a point in the field space, the coefficients ${a}_{i}$ and ${b}_{i}$ are "generally" well-behaved thermodynamic quantities and the ${f}_{\mathrm{ii}}$ , which are universal functions, have the following properties: ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is nondivergent if $i\ensuremath{\ge}3$, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{22}(q, \ensuremath{\omega})$ is weakly divergent, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is strongly divergent. A few typical examples are worked out to show that if this property is postulated, the Griffiths and Wheeler thermodynamic behavior is reproduced.A simple generalization of the Griffiths and Wheeler postulates for the thermodynamic behavior near a critical point is introduced to describe fluctuations in local dynamical variables. It is proposed that in the classical Hilbert space of the dynamical variables there are only two linearly independent vectors, related to the shape of the coexistence surface and the critical surface, which contain critical fluctuations. It is shown that, as a consequence of the affine structure of the field space, the above assumption leads to the following asymptotic property for the correlation function ${f}_{\mathrm{ab}}(q, \ensuremath{\omega})$ of any two local operators $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{a}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{b}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$: ${f}_{\mathrm{ab}}(q, \ensuremath{\omega})={a}_{1}(q, \ensuremath{\omega}){b}_{1}(q, \ensuremath{\omega}){f}_{11}(q, \ensuremath{\omega})+{a}_{2}(q, \ensuremath{\omega}){b}_{2}(q, \ensuremath{\omega}){f}_{22}(q, \ensuremath{\omega})+{a}_{3}(q, \ensuremath{\omega}){b}_{3}(q, \ensuremath{\omega}){f}_{33}(q, \ensuremath{\omega})+\dots{}$Here $q$ is the wave number, $\ensuremath{\omega}$ is a point in the field space, the coefficients ${a}_{i}$ and ${b}_{i}$ are "generally" well-behaved thermodynamic quantities and the ${f}_{\mathrm{ii}}$, which are universal functions, have the following properties: ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is nondivergent if $i\ensuremath{\ge}3$, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{22}(q, \ensuremath{\omega})$ is weakly divergent, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is strongly divergent. A few typical examples are worked out to show that if this property is postulated, the Griffiths and Wheeler thermodynamic behavior is reproduced.

Physical Review AJul 1, 19725 citations

On the theory of critical point density fluctuations

Michael Ellis Fisher

PhysicaFeb 1, 196252 citations

Correlation functions of order-parameter fluctuations in a Fermi system

Partha Goswami

Inverse Fourier transforms of dynamic correlation functions of order-parameter fluctuations in the singlet-spin-pairing and equal-spin-pairing states of an interacting Fermi system are calculated in a mean-field approximation. The behavior of these functions in the corresponding critical regions, i.e., in the neighborhood of a critical point for an anisotropic, singlet superfluid (and for a superconductor) and for an equal-spin-pairing-type, triplet superfluid, are examined. It is found that these functions do not conform to certain generalized homogeneity assumptions. The static counterparts of these functions are found to be nonsingular at corresponding critical points. A comparison of these findings with corresponding findings for a Bose system, reported in a previous work, highlights the role of Fermi statistics in the critical regions under consideration.Inverse Fourier transforms of dynamic correlation functions of order-parameter fluctuations in the singlet-spin-pairing and equal-spin-pairing states of an interacting Fermi system are calculated in a mean-field approximation. The behavior of these functions in the corresponding critical regions, i.e., in the neighborhood of a critical point for an anisotropic, singlet superfluid (and for a superconductor) and for an equal-spin-pairing-type, triplet superfluid, are examined. It is found that these functions do not conform to certain generalized homogeneity assumptions. The static counterparts of these functions are found to be nonsingular at corresponding critical points. A comparison of these findings with corresponding findings for a Bose system, reported in a previous work, highlights the role of Fermi statistics in the critical regions under consideration.

Physical review. B, Condensed matterFeb 1, 19871 citation

q

is the wave number, $\ensuremath{\omega}$ is a point in the field space, the coefficients

{a}_{i}

and

{b}_{i}

are "generally" well-behaved thermodynamic quantities and the

{f}_{\mathrm{ii}}

, which are universal functions, have the following properties: ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is nondivergent if $i\ensuremath{\ge}3$, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{22}(q, \ensuremath{\omega})$ is weakly divergent, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is strongly divergent. A few typical examples are worked out to show that if this property is postulated, the Griffiths and Wheeler thermodynamic behavior is reproduced.