Some Research in Ergodic Theory and Mathematical Problems of Statistical Mechanics in the Moscow University Department of Probability Theory

1990 ◽  
Vol 34 (1) ◽  
pp. 186-193
Author(s):  
V. A. Malyshev ◽  
Ya. G. Sinai
Keyword(s):  
Probability Theory ◽  
Statistical Mechanics ◽  
Ergodic Theory ◽  
Mathematical Problems ◽  
University Department

Introduction

2017 ◽  
Author(s):  
Jochen Rau
Keyword(s):  
Probability Theory ◽  
Phase Transitions ◽  
Statistical Mechanics ◽  
Conservation Laws ◽  
Law Of Large Numbers ◽  
Macroscopic Scale ◽  
Classical Probability ◽  
Large Numbers ◽  
Microscopic World ◽  
New Phenomena

Statistical mechanics concerns the transition from the microscopic to the macroscopic realm. On a macroscopic scale new phenomena arise that have no counterpart in the microscopic world. For example, macroscopic systems have a temperature; they might undergo phase transitions; and their dynamics may involve dissipation. How can such phenomena be explained? This chapter discusses the characteristic differences between the microscopic and macroscopic realms and lays out the basic challenge of statistical mechanics. It suggests how, in principle, this challenge can be tackled with the help of conservation laws and statistics. The chapter reviews some basic notions of classical probability theory. In particular, it discusses the law of large numbers and illustrates how, despite the indeterminacy of individual events, statistics can make highly accurate predictions about totals and averages.

Ergodic Theory in Statistical Mechanics

Physics Today ◽  
1966 ◽  
Vol 19 (6) ◽  
pp. 91-92
Author(s):  
I. E. Farquhar ◽  
R. B. Lindsay
Keyword(s):  
Statistical Mechanics ◽  
Ergodic Theory

scholarly journals Ergodic theorem, ergodic theory, and statistical mechanics

2015 ◽  
Vol 112 (7) ◽  
pp. 1907-1911 ◽  
Author(s):  
Calvin C. Moore
Keyword(s):  
Statistical Mechanics ◽  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Fundamental Problem ◽  
Ergodic Theorems ◽  
Von Neumann ◽  
Ergodic Hypothesis ◽  
Recent Developments ◽  
The Subject ◽  
Time Averages

This perspective highlights the mean ergodic theorem established by John von Neumann and the pointwise ergodic theorem established by George Birkhoff, proofs of which were published nearly simultaneously in PNAS in 1931 and 1932. These theorems were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60-y-old fundamental problem of the subject—namely, the rationale for the hypothesis that time averages can be set equal to phase averages. The evolution of this problem is traced from the origins of statistical mechanics and Boltzman's ergodic hypothesis to the Ehrenfests' quasi-ergodic hypothesis, and then to the ergodic theorems. We discuss communications between von Neumann and Birkhoff in the Fall of 1931 leading up to the publication of these papers and related issues of priority. These ergodic theorems initiated a new field of mathematical-research called ergodic theory that has thrived ever since, and we discuss some of recent developments in ergodic theory that are relevant for statistical mechanics.

scholarly journals ANALISIS VARIABEL PEMBELAJARAN BERDASARKAN TEORI REIGELUTH (Studi pada Mahasiswa Pendidikan Matematika FTIK IAIN Langsa)

2020 ◽  
Vol 3 (2) ◽  
pp. 104
Author(s):  
Srimuliati Srimuliati
Keyword(s):  
Probability Theory ◽  
Learning Outcomes ◽  
Descriptive Analysis ◽  
Learning Objectives ◽  
Story Problems ◽  
Study Program ◽  
Learning Conditions ◽  
Education Study ◽  
Mathematical Problems ◽  
Research Findings

Learning is an attempt to compare what behaviors might occur before individuals are placed in learning situations and what behaviors can be demonstrated after being given treatment. The purpose of this study was to analyze the learning variables according to teroi reigeluth which were divided into 3 things, namely (1) learning conditions, (2) learning methods and (3) learning outcomes. Subjects in this study were students of Mathematics Education Study Program FTIK IAIN Langsa in semester IV unit 1 who took courses in calculus and probability theory. By using descriptive analysis the results and research findings obtained in the form of learning conditions (learning objectives and characters, learner characters, and constraints in learning) for both courses namely calculus and opportunity theory give different results. Fundamental differences related to the two materials provide a finding that students are lacking in reasoning and sharpening their thinking power both through reading and discussing mathematical problems so that they are often wrong in interpreting story problems on occasion. Learning outcomes for both courses are in very good categories for calculus and good for probability theory.

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