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A Journey Through Complex Landscapes—Dedicated to Professor Giorgio Parisi to Celebrate the Nobel Prize & His 75th Birthday

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: closed (31 August 2024) | Viewed by 4894

Special Issue Editor

Dr. Giuseppe Florio

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Guest Editor
Dipartimento di Ingegneria Civile, Ambientale, del Territorio, Edile e di Chimica, Politecnico di Bari, I–70125 Bari, Italy
Interests: statistical mechanics; modeling of macromolecules and bio-inspired materials; quantum correlations

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is bringing together contributions and review articles about recent research activities in the field of complex systems. As a matter of fact, this subject has witnessed an enormous interest in the past years. The features of a complex system typically arise from interactions and competitions among the elementary constituents. As a consequence, the whole system can exhibit peculiar phenomena as nonlinearity, self-organization, and emergence. The field has seen the simultaneous development of new concepts and powerful analytical and numerical mathematical methods. These tools have been used to study models that can be applied to a large number of problems, ranging from collective phenomena in condensed matter physics and biology to climate changes, networks and economic systems.

We welcome articles about concepts and methods in statistical physics with an emphasis on systems with many degrees of freedom. We encourage to submit contributions devoted to analytical and numerical methods. Papers and reviews about spin glasses and applications to materials, soft matter and polymers are welcome. Topics can also include the use of mathematical methods and statistical physics in neural networks (for instance, with applications to machine learning). Papers can also address problems related to biological systems such as biopolymers, folding/unfolding phenomena, formation of structures, cellular mechanics and bioinformatics. Finally, we welcome papers related to quantum properties and phenomena such as quantum correlations in many-body systems and quantum many-body localization.  

Prof. Giorgio Parisi is a leading scientist in the development of methods for the study of complex systems that had large application in research fields involving statistical physics, condensed matter and spin glasses, mathematical physics, biology and collective phenomena. Moreover, Prof. Parisi has made fundamental contributions to the theory of elementary particles, field theory, the study of growth models and the application of stochastic resonance in the study of climatic phenomena. Prof. Parisi's work has earned him the Wolf Prize, the Boltzmann Medal, the Enrico Fermi Prize, the Dirac Medal and, finally, the Nobel Prize in Physics in 2021. This Special Issue is dedicated to him on the occasion of the Noble Prize and his 75th birthday.

Dr. Giuseppe Florio
Guest Editor

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Keywords

  • complex systems
  • statistical mechanics
  • spin glasses
  • many-body systems
  • neural networks
  • biological phenomena
  • collective phenomena
  • quantum correlations

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Published Papers (4 papers)

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Research

10 pages, 364 KiB  
Article
Kramers–Wannier Duality and Random-Bond Ising Model
by Chaoming Song
Entropy 2024, 26(8), 636; https://doi.org/10.3390/e26080636 - 27 Jul 2024
Viewed by 585
Abstract
We present a new combinatorial approach to the Ising model incorporating arbitrary bond weights on planar graphs. In contrast to existing methodologies, the exact free energy is expressed as the determinant of a set of ordered and disordered operators defined on a planar [...] Read more.
We present a new combinatorial approach to the Ising model incorporating arbitrary bond weights on planar graphs. In contrast to existing methodologies, the exact free energy is expressed as the determinant of a set of ordered and disordered operators defined on a planar graph and the corresponding dual graph, respectively, thereby explicitly demonstrating the Kramers–Wannier duality. The implications of our derived formula for the Random-Bond Ising Model are further elucidated. Full article
(This article belongs to the Special Issue A Journey Through Complex Landscapes—Dedicated to Professor Giorgio Parisi to Celebrate the Nobel Prize & His 75th Birthday)
►▼ Show Figures

Figure 1

Figure 1
<p>(<b>a</b>) The embedding of both <span class="html-italic">G</span> (red) and its dual <math display="inline"><semantics> <msup> <mi>G</mi> <mo>∗</mo> </msup> </semantics></math> (blue). The quadrilateral <span class="html-italic">q</span> is delineated by a vertex <span class="html-italic">v</span> and a neighboring dual vertex <math display="inline"><semantics> <msup> <mi>v</mi> <mo>∗</mo> </msup> </semantics></math>, along with their respective edges. The relationships <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mi>L</mi> </msub> <mo>+</mo> <msubsup> <mi>θ</mi> <mi>R</mi> <mo>∗</mo> </msubsup> <mo>=</mo> <msub> <mi>θ</mi> <mi>R</mi> </msub> <mo>+</mo> <msubsup> <mi>θ</mi> <mi>L</mi> <mo>∗</mo> </msubsup> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> are satisfied. (<b>b</b>) The local order and disorder operators, <math display="inline"><semantics> <msub> <mi>d</mi> <mi>v</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>d</mi> <mrow> <msup> <mi>v</mi> <mo>∗</mo> </msup> </mrow> <mo>†</mo> </msubsup> </semantics></math>, are applied to quadrilaterals, which are highlighted by different color regions. Each operator acts as a curl operator around the vertex <span class="html-italic">v</span> and the dual <math display="inline"><semantics> <msup> <mi>v</mi> <mo>∗</mo> </msup> </semantics></math>, respectively.</p> Full article ">Figure 2
<p>(<b>a</b>) The exterior angle <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>(</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>e</mi> <mo>)</mo> </mrow> </semantics></math> and the interior angle <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>(</mo> <mover accent="true"> <mi>e</mi> <mo stretchy="false">¯</mo> </mover> <mo>,</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </semantics></math> for the KW operator satisfy <math display="inline"><semantics> <mrow> <mi>α</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>,</mo> <mi>e</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mi>β</mi> <mrow> <mo>(</mo> <mover accent="true"> <mi>e</mi> <mo stretchy="false">¯</mo> </mover> <mo>,</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math>. (<b>b</b>) The angles between two neighboring edges attached to a quadrilateral satisfy <math display="inline"><semantics> <mrow> <mi>β</mi> <mrow> <mo>(</mo> <mover accent="true"> <mi>e</mi> <mo stretchy="false">¯</mo> </mover> <mo>,</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>β</mi> <mrow> <mo>(</mo> <mover accent="true"> <mi>e</mi> <mo stretchy="false">¯</mo> </mover> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mo>″</mo> </mrow> </msup> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>θ</mi> <msup> <mi>e</mi> <mo>′</mo> </msup> </msub> <mo>−</mo> <msub> <mi>θ</mi> <msup> <mi>e</mi> <mrow> <mo>″</mo> </mrow> </msup> </msub> </mrow> </semantics></math>.</p> Full article ">
11 pages, 318 KiB  
Article
Restoring the Fluctuation–Dissipation Theorem in Kardar–Parisi–Zhang Universality Class through a New Emergent Fractal Dimension
by Márcio S. Gomes-Filho, Pablo de Castro, Danilo B. Liarte and Fernando A. Oliveira
Entropy 2024, 26(3), 260; https://doi.org/10.3390/e26030260 - 14 Mar 2024
Cited by 1 | Viewed by 1179
Abstract
The Kardar–Parisi–Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth [...] Read more.
The Kardar–Parisi–Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth exponents; and the apparent absence of a fluctuation–dissipation theorem (FDT) for spatial dimension d>1. Notably, these questions were answered exactly only for 1+1 dimensions. In this work, we propose a new FDT valid for the KPZ problem in d+1 dimensions. This is achieved by rearranging terms and identifying a new correlated noise which we argue to be characterized by a fractal dimension dn. We present relations between the KPZ exponents and two emergent fractal dimensions, namely df, of the rough interface, and dn. Also, we simulate KPZ growth to obtain values for transient versions of the roughness exponent α, the surface fractal dimension df and, through our relations, the noise fractal dimension dn. Our results indicate that KPZ may have at least two fractal dimensions and that, within this proposal, an FDT is restored. Finally, we provide new insights into the old question about the upper critical dimension of the KPZ universality class. Full article
(This article belongs to the Special Issue A Journey Through Complex Landscapes—Dedicated to Professor Giorgio Parisi to Celebrate the Nobel Prize & His 75th Birthday)
►▼ Show Figures

Figure 1

Figure 1
<p>SS model in <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math> dimensions: The roughness exponent <math display="inline"><semantics> <mi>α</mi> </semantics></math> as a function of time t (in units of <math display="inline"><semantics> <msub> <mi>t</mi> <mo>×</mo> </msub> </semantics></math>) for a system of size <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>4096</mn> </mrow> </semantics></math> obtained from the correlation function (<a href="#FD14-entropy-26-00260" class="html-disp-formula">14</a>). The dashed line represents the stationary theoretically exact value for <math display="inline"><semantics> <mi>α</mi> </semantics></math>, i.e., 1/2.</p> Full article ">Figure 2
<p>Fractal dimensions <math display="inline"><semantics> <msub> <mi>d</mi> <mi>f</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>d</mi> <mi>n</mi> </msub> </semantics></math> as a function of time <span class="html-italic">t</span> for the SS model in <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math> dimensions. The dashed lines represent the stationary theoretical values for each fractal dimension (see text).</p> Full article ">Figure 3
<p>SS model in <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math> dimensions. (<b>a</b>): Fractal dimensions <math display="inline"><semantics> <msub> <mi>d</mi> <mi>f</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>d</mi> <mi>n</mi> </msub> </semantics></math> against time <span class="html-italic">t</span>. The dashed line represents the theoretical value for <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mi>n</mi> </msub> <mo>=</mo> <msub> <mi>d</mi> <mi>f</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>5</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </mrow> </semantics></math> (golden ration). (<b>b</b>): The difference between the fractal dimensions, <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mi>f</mi> </msub> <mo>−</mo> <msub> <mi>d</mi> <mi>n</mi> </msub> </mrow> </semantics></math>, as a function of time. The dashed line marks zero, whereas the horizontal solid line represents the average value, <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>0.0011</mn> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics></math>, obtained within the time interval from <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>5</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> </mrow> </semantics></math>. In the insets, we zoom into the stationary regime data.</p> Full article ">Figure 4
<p>SS model in <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math> dimensions: Fractal dimensions <math display="inline"><semantics> <msub> <mi>d</mi> <mi>f</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>d</mi> <mi>n</mi> </msub> </semantics></math> as a function of time <span class="html-italic">t</span>.</p> Full article ">
19 pages, 1151 KiB  
Article
Dynamic Phase Transition in 2D Ising Systems: Effect of Anisotropy and Defects
by Federico Ettori, Thibaud Coupé, Timothy J. Sluckin, Ezio Puppin and Paolo Biscari
Entropy 2024, 26(2), 120; https://doi.org/10.3390/e26020120 - 29 Jan 2024
Viewed by 1079
Abstract
We investigate the dynamic phase transition in two-dimensional Ising models whose equilibrium characteristics are influenced by either anisotropic interactions or quenched defects. The presence of anisotropy reduces the dynamical critical temperature, leading to the expected result that the critical temperature approaches zero in [...] Read more.
We investigate the dynamic phase transition in two-dimensional Ising models whose equilibrium characteristics are influenced by either anisotropic interactions or quenched defects. The presence of anisotropy reduces the dynamical critical temperature, leading to the expected result that the critical temperature approaches zero in the full-anisotropy limit. We show that a comprehensive understanding of the dynamic behavior of systems with quenched defects requires a generalized definition of the dynamic order parameter. By doing so, we demonstrate that the inclusion of quenched defects lowers the dynamic critical temperature as well, with a linear trend across the range of defect fractions considered. We also explore if and how it is possible to predict the dynamic behavior of specific magnetic systems with quenched randomness. Various geometric quantities, such as a defect potential index, the defect dipole moment, and the properties of the defect Delaunay triangulation, prove useful for this purpose. Full article
(This article belongs to the Special Issue A Journey Through Complex Landscapes—Dedicated to Professor Giorgio Parisi to Celebrate the Nobel Prize & His 75th Birthday)
►▼ Show Figures

Figure 1

Figure 1
<p>Time−dependent average magnetization (orange) in presence of an oscillating field (blue) for different choices of temperature for an isotropic, defect-less Ising system. <b>Left</b>: The magnetization follows the field, albeit with a time delay. The system is in its dynamically disordered phase (<math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>1.9</mn> <mspace width="0.166667em"/> <mi>J</mi> </mrow> </semantics></math>). <b>Right</b>: The magnetization does not reverse its sign, and the system is in its dynamically ordered phase (here, <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>1.5</mn> <mspace width="0.166667em"/> <mi>J</mi> </mrow> </semantics></math>). Simulations over a 2D square system with size <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math>, averaging 20 cycles for a field with <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mspace width="0.166667em"/> <mi>J</mi> </mrow> </semantics></math> and period <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mn>258</mn> </mrow> </semantics></math>. For these values of the parameters, the dynamic critical temperature is <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Θ</mi> <mi mathvariant="normal">c</mi> </msub> <mo>=</mo> <mn>1.8</mn> <mspace width="0.166667em"/> <mi>J</mi> </mrow> </semantics></math>. The reported magnetic field <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> is measured in units of <span class="html-italic">J</span>.</p> Full article ">Figure 2
<p>DPT in an anisotropic Ising system. <b>Top left</b>: Average magnetization per cycle as a function of temperature for heating/cooling experiments (no hysteresis effects). The external field amplitude is set to <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, the values chosen for the anisotropy <math display="inline"><semantics> <mi>λ</mi> </semantics></math> are displayed in the inset, and the points’ shapes correspond to different system sizes: <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> (*), 100 (+), 140 (∘), 180 (△), and 220 (◁). Finite-size effects are not relevant, as all points collapse into the same curves. <b>Top right</b>: Susceptibility curve as a function of temperature. The peak spots the critical temperature for the DPT. <b>Bottom</b>: Binder cumulant as a function of temperature for a fixed anisotropy <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>16</mn> <mi>π</mi> </mrow> </semantics></math>. The inset shows the separation of the curves for systems with different sizes in the vicinity of the critical temperature.</p> Full article ">Figure 3
<p>Dynamic critical temperature as a function of the inverse system size for the anisotropic Ising model. For each choice of anisotropy, a solid line is shown to guide the eye. For <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>≳</mo> <mn>60</mn> </mrow> </semantics></math>, no significant variation in the critical temperature is seen.</p> Full article ">Figure 4
<p>Critical temperature <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Θ</mi> <mi mathvariant="normal">c</mi> </msub> </semantics></math> for the DPT for a 2D anisotropic Ising model, as a function of the anisotropy coefficient <math display="inline"><semantics> <mi>λ</mi> </semantics></math>. The black dotted line represents the static critical temperature as derived in [<a href="#B38-entropy-26-00120" class="html-bibr">38</a>]. For each value of <math display="inline"><semantics> <msub> <mi>h</mi> <mn>0</mn> </msub> </semantics></math>, the factor <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>(</mo> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </semantics></math>, introduced in (<a href="#FD16-entropy-26-00120" class="html-disp-formula">16</a>) and reported in the right panel, has been computed by minimizing the residual error <math display="inline"><semantics> <mrow> <msub> <mo>∑</mo> <mi>λ</mi> </msub> <msup> <mfenced separators="" open="(" close=")"> <msub> <mi mathvariant="sans-serif">Θ</mi> <mi mathvariant="normal">c</mi> </msub> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>T</mi> <mi mathvariant="normal">c</mi> </msub> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> </mfenced> <mn>2</mn> </msup> </mrow> </semantics></math>. The continuous plot corresponds to the thermodynamical critical temperature, computed from the analytical solution obtained in the absence of an oscillating field.</p> Full article ">Figure 5
<p><b>Top Left</b>: Average magnetization per cycle as a function of temperature for a fixed external field amplitude <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, with three choices of the fraction of defects <span class="html-italic">f</span> and different system sizes: <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> (*), 100 (+), 180 (∘), 260 (△). <b>Top Right</b>: Dynamic susceptibility as a function of temperature. Their peaks locate the dynamic critical temperature. <b>Bottom</b>: Binder cumulant as a function of temperature for the system with defects <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>1</mn> <mo>%</mo> </mrow> </semantics></math>. The inset shows a zoomed-in region of the intersection of the curves.</p> Full article ">Figure 6
<p>Dynamic critical temperature as a function of the inverse system size. Different choices of the fraction of defects are displayed for the case <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>. The dynamic critical temperature is determined in the thermodynamic limit by using a linear interpolation, in accordance with [<a href="#B24-entropy-26-00120" class="html-bibr">24</a>].</p> Full article ">Figure 7
<p><b>Left</b>: Dynamical critical temperature as a function of the defect fraction. The linear extrapolation for <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>→</mo> <mn>0</mn> </mrow> </semantics></math> in the <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> case coincides with the defect-free critical temperature derived in [<a href="#B24-entropy-26-00120" class="html-bibr">24</a>]. <b>Right</b>: Dynamical critical temperature as a function of the external field amplitude. The linear extrapolation for <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>→</mo> <mn>0</mn> </mrow> </semantics></math> is compatible with the Curie temperature for all the data series.</p> Full article ">Figure 8
<p>Correlation between the potential <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> and the local average magnetization per cycle <math display="inline"><semantics> <msub> <mi>Q</mi> <mi>i</mi> </msub> </semantics></math> at the dynamic critical temperature for a system with <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>260</mn> </mrow> </semantics></math>. The solid line represents the average value, while dashed lines delimit the standard deviation of the potential distribution of sites exhibiting the same value of <math display="inline"><semantics> <msub> <mi>Q</mi> <mi>i</mi> </msub> </semantics></math>.</p> Full article ">Figure 9
<p>Sequence of defect configurations with increasing values of the potential index for a system with <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> and a defect fraction <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>2</mn> <mo>%</mo> </mrow> </semantics></math>. In the top row (from left to right): <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>def</mi> </msub> <mo>=</mo> <mn>0.049</mn> </mrow> </semantics></math>, 0.065, and 0.13. In the bottom row (again, from left to right): <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>def</mi> </msub> <mo>=</mo> <mn>0.26</mn> </mrow> </semantics></math>, 0.52, and 0.98. The color code of the plots is determined by the local potential <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math>. The color scale has been adjusted across the images; otherwise, the upper configurations would have all appeared predominantly orange. Yellow (black) squares represent positive (negative) defects.</p> Full article ">Figure 10
<p>Log–log plot of the average potential index <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mi>def</mi> </msub> </semantics></math> (<b>left</b>) and its standard deviation <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>ϕ</mi> </msub> </semantics></math> (<b>right</b>), as a function of both the system size <span class="html-italic">L</span> and the defect fraction <span class="html-italic">f</span>. The two insets show that data associated with different values of <span class="html-italic">f</span> collapse into a universal curve by plotting <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>def</mi> </msub> <msup> <mi>f</mi> <mrow> <mn>0.3</mn> </mrow> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>ϕ</mi> </msub> <msup> <mi>f</mi> <mrow> <mn>0.3</mn> </mrow> </msup> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>Graphical representation of the Delaunay triangulation for the same realizations of defects shown in <a href="#entropy-26-00120-f009" class="html-fig">Figure 9</a>. We obtain, respectively, the values for the configuration index: <math display="inline"><semantics> <mrow> <mi mathvariant="script">A</mi> <mo>=</mo> <mn>1.42</mn> </mrow> </semantics></math>, 1.38, and 1.46 for the top row, and <math display="inline"><semantics> <mrow> <mi mathvariant="script">A</mi> <mo>=</mo> <mn>1.76</mn> </mrow> </semantics></math>, 2.17, and 1.46 for the bottom row. The colors specify triangles with different values of <math display="inline"><semantics> <msub> <mi>e</mi> <mi>i</mi> </msub> </semantics></math>: dark blue for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mrow> </semantics></math>, light blue for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>, orange for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and red for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>Correlations between dynamic critical temperature (<b>left column</b>) and dynamic susceptibility peak (<b>right column</b>) and two geometric parameters: the total potential (<b>top row</b>) and the normalized area index (<b>bottom row</b>). For the total potential, each point represents the average over 10 realizations of defects with the specified target potential. The error bar represents the standard deviation. For the normalized area index, all 200 realizations have been analyzed, with a data binning procedure of 18 bins. The point represents the average dynamical property for each bin, and the error bar is its standard deviation. In all the panels, the orange window locates the interval of values where most random configurations can be found, as the position and width of the window are determined by the average and standard deviation of the geometric quantity as computed among random configurations. The thermodynamic properties were obtained by following 20 systems across <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>5000</mn> </mrow> </semantics></math> cycles under an oscillating magnetic field with intensity <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> for a system of size <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>.</p> Full article ">
15 pages, 331 KiB  
Article
The Onset of Parisi’s Complexity in a Mismatched Inference Problem
by Francesco Camilli, Pierluigi Contucci and Emanuele Mingione
Entropy 2024, 26(1), 42; https://doi.org/10.3390/e26010042 - 30 Dec 2023
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Abstract
We show that a statistical mechanics model where both the Sherringhton–Kirkpatrick and Hopfield Hamiltonians appear, which is equivalent to a high-dimensional mismatched inference problem, is described by a replica symmetry-breaking Parisi solution. Full article
(This article belongs to the Special Issue A Journey Through Complex Landscapes—Dedicated to Professor Giorgio Parisi to Celebrate the Nobel Prize & His 75th Birthday)
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