摘要翻译:
我们构造了一个具有多尺度、平移不变、对数相关的n维高斯景观,并研究了单个粒子在该环境中的统计力学。在高维N>>1的极限下,利用复制技巧和Parisi的层次分析法,精确地计算了系统的自由能和重叠函数。在热力学极限下,我们恢复了德里达广义随机能量模型(GREM)的最一般版本。低温行为主要取决于景观建设中涉及的长度尺度光谱。如果后者由K个离散值组成,则系统用K步复制对称破缺解来表征。我们认为我们的构造实际上在任何有限的空间维度中都是有效的。我们讨论了我们的结果对于描述相关Boltzmann-Gibbs测度多重分形的奇异谱的含义。最后,我们讨论了几个推广和开放问题,在这样的景观中的动力学和广义多重分形随机游动的构造。
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英文标题:
《Statistical mechanics of a single particle in a multiscale random
potential: Parisi landscapes in finite dimensional Euclidean spaces》
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作者:
Yan V Fyodorov and Jean-Philippe Bouchaud
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最新提交年份:
2008
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分类信息:
一级分类:Physics 物理学
二级分类:Disordered Systems and Neural Networks 无序系统与神经网络
分类描述:Glasses and spin glasses; properties of random, aperiodic and quasiperiodic systems; transport in disordered media; localization; phenomena mediated by defects and disorder; neural networks
眼镜和旋转眼镜;随机、非周期和准周期系统的性质;无序介质中的传输;本地化;由缺陷和无序介导的现象;神经网络
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一级分类:Physics 物理学
二级分类:Statistical Mechanics 统计力学
分类描述:Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变,热力学,场论,非平衡现象,重整化群和标度,可积模型,湍流
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英文摘要:
We construct a N-dimensional Gaussian landscape with multiscale, translation invariant, logarithmic correlations and investigate the statistical mechanics of a single particle in this environment. In the limit of high dimension N>>1 the free energy of the system and overlap function are calculated exactly using the replica trick and Parisi's hierarchical ansatz. In the thermodynamic limit, we recover the most general version of the Derrida's Generalized Random Energy Model (GREM). The low-temperature behaviour depends essentially on the spectrum of length scales involved in the construction of the landscape. If the latter consists of K discrete values, the system is characterized by a K-step Replica Symmetry Breaking solution. We argue that our construction is in fact valid in any finite spatial dimensions $N\ge 1$. We discuss implications of our results for the singularity spectrum describing multifractality of the associated Boltzmann-Gibbs measure. Finally we discuss several generalisations and open problems, the dynamics in such a landscape and the construction of a Generalized Multifractal Random Walk.
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PDF链接:
https://arxiv.org/pdf/711.4006