Finite size distribution and partition functions of Gaussian chains: maximum entropy approach

1992 ◽  
Vol 25 (25) ◽  
pp. 6891-6896 ◽  
Author(s):  
R. G. Winkler ◽  
P. Reineker
Keyword(s):  
Maximum Entropy ◽  
Size Distribution ◽  
Finite Size ◽  
Partition Functions

scholarly journals Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 895
Author(s):  
Nickolay Izmailian ◽  
Ralph Kenna
Keyword(s):  
Generating Functions ◽  
Spanning Trees ◽  
Central Charge ◽  
Finite Size ◽  
Universality Class ◽  
Partition Functions ◽  
Free Energies ◽  
Conformal Theory ◽  
Alternative Means ◽  
The Individual

The concept of universality is a cornerstone of theories of critical phenomena. It is very well understood in most systems, especially in the thermodynamic limit. Finite-size systems present additional challenges. Even in low dimensions, universality of the edge and corner contributions to free energies and response functions is less investigated and less well understood. In particular, the question arises of how universality is maintained in correction-to-scaling in systems of the same universality class but with very different corner geometries. Two-dimensional geometries deliver the simplest such examples that can be constructed with and without corners. To investigate how the presence and absence of corners manifest universality, we analyze the spanning tree generating function on two different finite systems, namely the cobweb and fan networks. The corner free energies of these configurations have stimulated significant interest precisely because of expectations regarding their universal properties and we address how this can be delivered given that the finite-size cobweb has no corners while the fan has four. To answer, we appeal to the Ivashkevich–Izmailian–Hu approach which unifies the generating functions of distinct networks in terms of a single partition function with twisted boundary conditions. This unified approach shows that the contributions to the individual corner free energies of the fan network sum to zero so that it precisely matches that of the web. It therefore also matches conformal theory (in which the central charge is found to be c = − 2 ) and finite-size scaling predictions. Correspondence in each case with results established by alternative means for both networks verifies the soundness of the Ivashkevich–Izmailian–Hu algorithm. Its broad range of usefulness is demonstrated by its application to hitherto unsolved problems—namely the exact asymptotic expansions of the logarithms of the generating functions and the conformal partition functions for fan and cobweb geometries. We also investigate strip geometries, again confirming the predictions of conformal field theory. Thus, the resolution of a universality puzzle demonstrates the power of the algorithm and opens up new applications in the future.

Partition functions and finite-size scalings of Ising model on helical tori

2006 ◽  
Vol 73 (5) ◽  
Author(s):  
Tsong-Ming Liaw ◽  
Ming-Chang Huang ◽  
Yen-Liang Chou ◽  
Simon Lin ◽  
Feng-Yin Li
Keyword(s):  
Ising Model ◽  
Finite Size ◽  
Partition Functions

scholarly journals PARTITION FUNCTIONS OF MATRIX MODELS: FIRST SPECIAL FUNCTIONS OF STRING THEORY

2004 ◽  
Vol 19 (24) ◽  
pp. 4127-4163 ◽  
Author(s):  
A. ALEXANDROV ◽  
A. MOROZOV ◽  
A. MIRONOV
Keyword(s):  
String Theory ◽  
Partition Function ◽  
Matrix Model ◽  
Special Functions ◽  
Finite Size ◽  
Building Blocks ◽  
Partition Functions ◽  
Virasoro Constraints ◽  
Elementary Building

Even though matrix model partition functions do not exhaust the entire set of τ-functions relevant for string theory, they seem to be elementary building blocks for many others and they seem to properly capture the fundamental symplicial nature of quantum gravity and string theory. We propose to consider matrix model partition functions as new special functions. Here we restrict our consideration to the finite-size Hermitian 1-matrix model and concentrate mostly on its phase/branch structure arising when the partition function is considered as a D-module. We discuss the role of the CIV–DV prepotential (as generating a possible basis in the linear space of solutions to the Virasoro constraints, but with a lack of understanding of why and how this basis is distinguished).

scholarly journals Using the Semantic Information G Measure to Explain and Extend Rate-Distortion Functions and Maximum Entropy Distributions

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1050
Author(s):  
Chenguang Lu
Keyword(s):  
Machine Learning ◽  
Mutual Information ◽  
Data Compression ◽  
Maximum Entropy ◽  
Semantic Information ◽  
Rate Distortion ◽  
Distortion Function ◽  
Partition Functions ◽  
Statistical Probability ◽  
Distortion Functions

In the rate-distortion function and the Maximum Entropy (ME) method, Minimum Mutual Information (MMI) distributions and ME distributions are expressed by Bayes-like formulas, including Negative Exponential Functions (NEFs) and partition functions. Why do these non-probability functions exist in Bayes-like formulas? On the other hand, the rate-distortion function has three disadvantages: (1) the distortion function is subjectively defined; (2) the definition of the distortion function between instances and labels is often difficult; (3) it cannot be used for data compression according to the labels’ semantic meanings. The author has proposed using the semantic information G measure with both statistical probability and logical probability before. We can now explain NEFs as truth functions, partition functions as logical probabilities, Bayes-like formulas as semantic Bayes’ formulas, MMI as Semantic Mutual Information (SMI), and ME as extreme ME minus SMI. In overcoming the above disadvantages, this paper sets up the relationship between truth functions and distortion functions, obtains truth functions from samples by machine learning, and constructs constraint conditions with truth functions to extend rate-distortion functions. Two examples are used to help readers understand the MMI iteration and to support the theoretical results. Using truth functions and the semantic information G measure, we can combine machine learning and data compression, including semantic compression. We need further studies to explore general data compression and recovery, according to the semantic meaning.

A SIMPLIFIED MAXIMUM-ENTROPY-BASED DROP SIZE DISTRIBUTION

1993 ◽  
Vol 3 (3) ◽  
pp. 291-310 ◽  
Author(s):  
Mehran Ahmadi ◽  
R. W. Sellens
Keyword(s):  
Maximum Entropy ◽  
Size Distribution ◽  
Drop Size ◽  
Drop Size Distribution
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