Bare essentials of statistical mechanics Atoms are examples of many-particle systems, but atoms are extraordinarily simpler than macroscopic systems consisting of 1020-1030 atoms. Despite their great size, many properties of macroscopic systems depend intimately on the microscopic behavior of their microscopic constituents. The proper quantum mechanical description of an N -particle system is a wavefunction that depends on 3N coordinates (3 ways of moving, in general, for every particle) and 4N quantum numbers (3 motional quantum numbers and 1 spin quantum number for every particle). (If the “particles” are molecules there might be additional quantum numbers describing rotations and internal vibrations.) The latter is referred to as a microstate. In a macroscopic solid or liquid there are about 1023 atoms per cm3; in a gas there are about 1020. Thus, the amount of information required to specify the wavefunction for such a system is scandalously large. A fundamental theory of macroscopic systems cannot be based on all of the microscopic details. Instead, a new way of thinking is required. This is known as statistical mechanics.
Bare essentials of statistical mechanics Atoms are examples of many-particle systems, but atoms are extraordinarily simpler than macroscopic systems consisting of 1020-1030 atoms. Despite their great size, many properties of macroscopic systems depend intimately on the microscopic behavior of their microscopic constituents. The proper quantum mechanical description of an N -particle system is a wavefunction that depends on 3N coordinates (3 ways of moving, in general, for every particle) and 4N quantum numbers (3 motional quantum numbers and 1 spin quantum number for every particle). (If the “particles” are molecules there might be additional quantum numbers describing rotations and internal vibrations.) The latter is referred to as a microstate. In a macroscopic solid or liquid there are about 1023 atoms per cm3; in a gas there are about 1020. Thus, the amount of information required to specify the wavefunction for such a system is scandalously large. A fundamental theory of macroscopic systems cannot be based on all of the microscopic details. Instead, a new way of thinking is required. This is known as statistical mechanics.
The basic concepts of statistical mechanics and its application to many-body systems, phase transitions and critical phenomena will be discussed briefly in this chapter. We will be using these ideas frequently in the subsequent lectures. Statistical mechanics describes the thermodynamic behaviour of macroscopic systems from the laws which govern the behaviour of the constituent elements at the microscopic level. The microscopic elements can be atoms, molecules, dipole moments or magnetic moments, etc. A macroscopic system is generally composed of a large number of these elements (of the order of Avogadro number N A ≈ 6.022 × 1023 per mole). Each element may have a large number of internal degrees of freedom associated with different types of motion such as translation, rotation, vibration etc. The constituent elements may interact with the external field applied to the system. There can also be very complex interaction among the constituent elements. The macroscopic properties of a system in the thermodynamic limit is thus determined by the properties of the constituent molecules, their interaction with external field as well as interaction among themselves. For a system of N elements, volume V and density ρ, the thermodynamic limit is defined as lim N → ∞, lim V → ∞ keeping ρ = N/V finite. In this limit, the extensive properties of the system become directly proportional to the size of the system (N or V), while the intensive properties become independent of the size of the system.
Introduction In our treatment of quantum mechanics we discussed the notion of the state of a given system. The state was specified by giving the eigenvalues of a maximal set of mutually commuting observables. This represented the most information we could have about a given system in principle. In practice, of course, we do not really have all of the necessary information about the eigenvalues available. No experimentalist sets out with the primary intention of obtaining results which correspond to a maximal set of mutually compatible observables for any macroscopic system, such as a gas. Even a very small volume of gas contains an enormous number of particles. The quantum mechanical state would require us to know a great deal of information about the internal states of the electrons and nuclei of the gas molecules. The experimentalist may only be concerned with gross properties such as the pressure, internal energy, or specific heat for example. Most of the detailed information about the detailed state of the system is neither observed, nor of direct interest. Instead we must content ourselves with a description of the average, or most likely, behaviour of the system in the probabilistic sense. The usual technique for doing this consists of the introduction of an ensemble, meaning a very large collection of systems whose measured macroscopic properties are the same as those for the system we are interested in, but whose unobserved properties can vary.
Many-particle systems play a fundamental role in physics where we encounter systems of molecules or systems of elementary particles in quantum field theory. Many-particle theories provide the basis for understanding the macroscopic behaviour of vast assemblies of interacting particles starting from their microscopic properties and leads to descriptions of entropy, statistical mechanics and fluid mechanics. Covering a wide range of topics and including more than 400 problems and solutions, this book is an ideal resource for students specialising in many-particle systems. Key features • Several hundred problems with solutions in each volume. • Problems range from basic introductory to advanced level. • Features worked examples as well as solutions to many problems. • Includes key concepts. • Covers essential complete physics course.
From the outset statistical mechanics will be framed in the language of quantum theory. The typical macroscopic system is composed of multiple constituents, and hence described in some many-particle Hilbert space. In general, not much is known about such a system, certainly not the precise preparation of all its microscopic details. Thus, its description requires a more general notion of a quantum state, a so-called mixed state. This chapter begins with a brief review of the basic axioms of quantum theory regarding observables, pure states, measurements, and time evolution. Particular attention is paid to the use of projection operators and to the most elementary quantum system, a two-level system. The chapter then motivates the introduction of mixed states and examines in detail their mathematical representation and properties. It also dwells on the description of composite systems, introducing, in particular, the notions of statistical independence and correlations.
Liquids and solids, atoms and molecules, nuclei—all these clearly are interacting many-body systems. Even a nucleon may be regarded as a manyparticle system, not just because it is now known to consist of three quarks interacting via gluons, but because of the possibility in quantum field theory of virtual excitation of many particles from the vacuum. in the table page 56 we list some of the many-particle systems we encounter in the physical world at length scales that range from a few centimeters to a few fermis. Many of these systems exhibit phenomena—superconductivity in solids and fission in nuclei, for example—whose understanding does not follow immediately from knowledge of the constituents of the system and the interactions among the constituents, but requires new concepts and ideas. Many-body physics is the branch of theoretical physics that studies the new phenomena or “emergent properties” that arise from interactions among “elementary” constituents of a many-particle system and provides means and methods for carrying out precise calculations of such characteristic properties of these systems as may be compared with experimental results to verify hypotheses about the nature of the constituents and their interactions.
Envisioned by Richard Feynman in the early 1980s, quantum simulation has received dramatic impetus thanks to the development of a variety of plateforms able to emulate a wide class of quantum Hamiltonians.During the past decade, most of the quantum simulators have implemented rather well-known models, hence permitting a direct comparison with theoretical calculations and a precise benchmarking of their reliability.The field has now reached a maturity such that one can address difficult problems, which cannot be solved efficiently using classical algorithms.These advances provide unprecedented opportunities to explore previously unreachable fields, test theoretical predictions, and inspire novel approaches.This contribution is an elementary introduction to quantum simulation.We discuss the challenges, define both digital and analog quantum simulators, and list the demanding conditions they require.We also provide a brief account of the contributions gathered in the dossier on Quantum simulation in the Compte rendus Physique of the French Academy of Sciences [1][2][3][4][5][6].The latter completes excellent reviews that appeared previously, see for instance . Universal models and the role of simulations in many-body physicsUnderstanding the behavior of macroscopic quantum systems is a major challenge of modern physics.The basic laws of low-energy physics are by now quite well known at the microscopic, say atomic, scale.Conversely, many fundamental questions remain open, and even debated, about the collective dynamics at the macroscopic scale.By "macroscopic scale", here we mean systems made up of a huge number of constituents, or degrees of freedom, say 10 6 , 10 23 , or even more, as relevant in condensed matter physics or in astrophysics, for instance.Such huge systems cannot be treated exactly, be it a the classical level and, even worse, at the quantum level.Yet, it is the main outcome of the thermodynamic approach that the collective behavior of a macroscopic system can drastically differ from that of its elementary constituents.For instance, the elementary interactions between the H 2 O molecules are fundamentally unchanged when a water bucket turns from the liquid phase to the solid phase at zero degree Celsius.Similarly, the interactions between the microscopic magnetic moments do not show any brutal change when a magnetic material gets magnetized underneath the Curie temperature.Hence, the dramatic effects observed in macroscopic systems are governed by large-scale instabilities, without obvious counterparts at the microscopic level.This observation takes a universal character, summarized in the celebrated motto "More is Different" [15].Such so-called emerging phenomena also appear in quantum systems, where new effects that are impossible in the classical world show up below some critical temperature or at zero temperature when some interaction parameter passes through a critical value.Celebrated examples include the superfluid transition in helium or super-conducting transitions and other metal-insulator transitions in electronic systems.Strikingly enough at first sight, while emerging phenomena only appear in very large scale macroscopic systems, their germs are contained in the mutual interactions of their elementary constituents.Hence, local two-body interactions are sufficient to explain a huge class of phase transitions, such as the liquid-solid and magnetic transitions mentioned above.The Boltzmann statistical approach proved particularly successful in describing this connection between the microscopic and macroscopic scales.To understand emerging phenomena on grounds as fundamental as possible, the programme is now well established: One tries to identify the basic microscopic terms that seem to be relevant and disregards all the other microscopic details.One then elaborates a model, as generic as possible, likely to reproduce the main experimental observations.The most usual examples are the Ising and Heisenberg models for magnetic transitions, or the Hubbard model for metal-insulator transitions [16][17][18].Then all is left to do is to check that the phenomenon of interest indeed emerges from the dynamics of the simplified model.The realization of this programme is nothing but a simulation.It consists in building up a simplified system that mimics the main properties of a real system.