*Stefan Adams*

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0007
- Subject:
- Mathematics, Probability / Statistics, Analysis

Motivated by the Bose gas, this chapter introduces certain combinatorial structures. It analyses the asymptotic behaviour of empirical shape measures and of empirical path measures of N Brownian ...
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Motivated by the Bose gas, this chapter introduces certain combinatorial structures. It analyses the asymptotic behaviour of empirical shape measures and of empirical path measures of N Brownian motions with large deviations techniques. The rate functions are given as variational problems that are analysed. A symmetrized system of Brownian motions is highly correlated and has to be formulated such that standard techniques can be applied. The chapter reviews a novel spatial and a novel cycle structure approach for the symmetrized distributions of the empirical path measures. The cycle structure leads to a proof of a phase transition in the mean path measure.Less

Motivated by the Bose gas, this chapter introduces certain combinatorial structures. It analyses the asymptotic behaviour of empirical shape measures and of empirical path measures of *N* Brownian motions with large deviations techniques. The rate functions are given as variational problems that are analysed. A symmetrized system of Brownian motions is highly correlated and has to be formulated such that standard techniques can be applied. The chapter reviews a novel spatial and a novel cycle structure approach for the symmetrized distributions of the empirical path measures. The cycle structure leads to a proof of a phase transition in the mean path measure.

*Stefan Adams and Wolfgang König*

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0008
- Subject:
- Mathematics, Probability / Statistics, Analysis

Bose–Einstein condensation predicts that, under certain conditions (in particular extremely low temperature), all particles will condense into one state. Some of the physical background is surveyed ...
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Bose–Einstein condensation predicts that, under certain conditions (in particular extremely low temperature), all particles will condense into one state. Some of the physical background is surveyed in this chapter. The Gross–Pitaevskii approximation for dilute systems is also discussed. Variational problems appear here naturally, as the quantum mechanical ground state is of interest. In connection with positive temperature, related probabilistic models, based on interacting Brownian motions in a trapping potential, are introduced. Again, large deviation techniques are used to determine the mean occupation measure, both for vanishing temperature and large particle number.Less

Bose–Einstein condensation predicts that, under certain conditions (in particular extremely low temperature), all particles will condense into one state. Some of the physical background is surveyed in this chapter. The Gross–Pitaevskii approximation for dilute systems is also discussed. Variational problems appear here naturally, as the quantum mechanical ground state is of interest. In connection with positive temperature, related probabilistic models, based on interacting Brownian motions in a trapping potential, are introduced. Again, large deviation techniques are used to determine the mean occupation measure, both for vanishing temperature and large particle number.

*Timo Seppäläinen*

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0001
- Subject:
- Mathematics, Probability / Statistics, Analysis

This chapter discusses random growth models describing the evolution of an interface in the plane. For specific models, three basic questions are discussed. First, under appropriate scaling, what is ...
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This chapter discusses random growth models describing the evolution of an interface in the plane. For specific models, three basic questions are discussed. First, under appropriate scaling, what is the limiting shape of the interface and what is the partial differential equation governing its evolution? Second, how can random fluctuations around the limit behaviour be described? Third, how can atypical behaviour be characterized? The power of probabilistic tools is demonstrated by employing laws of large numbers, central limit theorems, and large deviation techniques to answer these questions, respectively.Less

This chapter discusses random growth models describing the evolution of an interface in the plane. For specific models, three basic questions are discussed. First, under appropriate scaling, what is the limiting shape of the interface and what is the partial differential equation governing its evolution? Second, how can random fluctuations around the limit behaviour be described? Third, how can atypical behaviour be characterized? The power of probabilistic tools is demonstrated by employing laws of large numbers, central limit theorems, and large deviation techniques to answer these questions, respectively.