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106-422: In physics , statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics , its applications include many problems in the fields of physics, biology , chemistry , neuroscience , computer science , information theory and sociology . Its main purpose

212-499: A Platonist by Stephen Hawking , a view Penrose discusses in his book, The Road to Reality . Hawking referred to himself as an "unashamed reductionist" and took issue with Penrose's views. Mathematics provides a compact and exact language used to describe the order in nature. This was noted and advocated by Pythagoras , Plato , Galileo, and Newton. Some theorists, like Hilary Putnam and Penelope Maddy , hold that logical truths, and therefore mathematical reasoning, depend on

318-488: A frame of reference that is in motion with respect to an observer; the special theory of relativity is concerned with motion in the absence of gravitational fields and the general theory of relativity with motion and its connection with gravitation . Both quantum theory and the theory of relativity find applications in many areas of modern physics. While physics itself aims to discover universal laws, its theories lie in explicit domains of applicability. Loosely speaking,

424-601: A few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level. Many physical phenomena involve quasi-thermodynamic processes out of equilibrium, for example: All of these processes occur over time with characteristic rates. These rates are important in engineering. The field of non-equilibrium statistical mechanics

530-453: A finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics. For systems containing many particles (the thermodynamic limit ), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles

636-420: A hard-to-find physical meaning. The final mathematical solution has an easier-to-find meaning, because it is what the solver is looking for. Physics is a branch of fundamental science (also called basic science). Physics is also called " the fundamental science" because all branches of natural science including chemistry, astronomy, geology, and biology are constrained by laws of physics. Similarly, chemistry

742-422: A role in materials science, nuclear physics, astrophysics, chemistry, biology and medicine (e.g. study of the spread of infectious diseases). Analytical and computational techniques derived from statistical physics of disordered systems, can be extended to large-scale problems, including machine learning, e.g., to analyze the weight space of deep neural networks . Statistical physics is thus finding applications in

848-641: A simpler view of the subject." He had been working on this topic for some time, at least as early as 1884 when he produced a paper (now lost except for its abstract) on the topic of statistical mechanics. Gibbs' book simplified statistical mechanics into a treatise of 207 pages. At the same time, Gibbs fully generalized and expanded statistical mechanics into the form in which it is known today. Gibbs showed how statistical mechanics could be used even to extend thermodynamics beyond classical thermodynamics, to systems of any number of degrees of freedom (including microscopic systems) and non- extensive systems. At

954-465: A specific practical application as a goal, other than the deeper insight into the phenomema themselves. Applied physics is a general term for physics research and development that is intended for a particular use. An applied physics curriculum usually contains a few classes in an applied discipline, like geology or electrical engineering. It usually differs from engineering in that an applied physicist may not be designing something in particular, but rather

1060-426: A speed much less than the speed of light. These theories continue to be areas of active research today. Chaos theory , an aspect of classical mechanics, was discovered in the 20th century, three centuries after the original formulation of classical mechanics by Newton (1642–1727). These central theories are important tools for research into more specialized topics, and any physicist, regardless of their specialization,

1166-407: A state with a balance of forces that has ceased to evolve.) The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems. The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics)

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1272-399: A subfield of mechanics , is used in the building of bridges and other static structures. The understanding and use of acoustics results in sound control and better concert halls; similarly, the use of optics creates better optical devices. An understanding of physics makes for more realistic flight simulators , video games, and movies, and is often critical in forensic investigations. With

1378-481: A surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion. The founding of the field of statistical mechanics is generally credited to three physicists: In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius , Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave

1484-616: A term derived from φύσις ( phúsis 'origin, nature, property'). Astronomy is one of the oldest natural sciences . Early civilizations dating before 3000 BCE, such as the Sumerians , ancient Egyptians , and the Indus Valley Civilisation , had a predictive knowledge and a basic awareness of the motions of the Sun, Moon, and stars. The stars and planets, believed to represent gods, were often worshipped. While

1590-413: Is clear-cut, but not always obvious. For example, mathematical physics is the application of mathematics in physics. Its methods are mathematical, but its subject is physical. The problems in this field start with a " mathematical model of a physical situation " (system) and a "mathematical description of a physical law" that will be applied to that system. Every mathematical statement used for solving has

1696-410: Is completely erroneous, and our view may be corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a very small one. And so, if

1802-419: Is concerned with bodies acted on by forces and bodies in motion and may be divided into statics (study of the forces on a body or bodies not subject to an acceleration), kinematics (study of motion without regard to its causes), and dynamics (study of motion and the forces that affect it); mechanics may also be divided into solid mechanics and fluid mechanics (known together as continuum mechanics ),

1908-400: Is concerned with the most basic units of matter; this branch of physics is also known as high-energy physics because of the extremely high energies necessary to produce many types of particles in particle accelerators . On this scale, ordinary, commonsensical notions of space, time, matter, and energy are no longer valid. The two chief theories of modern physics present a different picture of

2014-499: Is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.) In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent,

2120-425: Is expected to be literate in them. These include classical mechanics, quantum mechanics, thermodynamics and statistical mechanics , electromagnetism , and special relativity. Classical physics includes the traditional branches and topics that were recognized and well-developed before the beginning of the 20th century—classical mechanics, acoustics , optics , thermodynamics, and electromagnetism. Classical mechanics

2226-511: Is firmly entrenched. Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics , a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous. Gibbs' methods were initially derived in the framework classical mechanics , however they were of such generality that they were found to adapt easily to

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2332-429: Is generally concerned with matter and energy on the normal scale of observation, while much of modern physics is concerned with the behavior of matter and energy under extreme conditions or on a very large or very small scale. For example, atomic and nuclear physics study matter on the smallest scale at which chemical elements can be identified. The physics of elementary particles is on an even smaller scale since it

2438-417: Is however a disconnect between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and

2544-593: Is often called the central science because of its role in linking the physical sciences. For example, chemistry studies properties, structures, and reactions of matter (chemistry's focus on the molecular and atomic scale distinguishes it from physics ). Structures are formed because particles exert electrical forces on each other, properties include physical characteristics of given substances, and reactions are bound by laws of physics, like conservation of energy , mass , and charge . Fundamental physics seeks to better explain and understand phenomena in all spheres, without

2650-506: Is possible only in discrete steps proportional to their frequency. This, along with the photoelectric effect and a complete theory predicting discrete energy levels of electron orbitals , led to the theory of quantum mechanics improving on classical physics at very small scales. Quantum mechanics would come to be pioneered by Werner Heisenberg , Erwin Schrödinger and Paul Dirac . From this early work, and work in related fields,

2756-449: Is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics. Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections. One approach to non-equilibrium statistical mechanics

2862-469: Is primarily concerned with thermodynamic equilibrium , statistical mechanics has been applied in non-equilibrium statistical mechanics to the issues of microscopically modeling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions and flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study

2968-400: Is provided by quantum logic . Physics Physics is the scientific study of matter , its fundamental constituents , its motion and behavior through space and time , and the related entities of energy and force . Physics is one of the most fundamental scientific disciplines. A scientist who specializes in the field of physics is called a physicist . Physics is one of

3074-400: Is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium . Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. (By contrast, mechanical equilibrium is

3180-482: Is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics , a field for which it was successful in explaining macroscopic physical properties—such as temperature , pressure , and heat capacity —in terms of microscopic parameters that fluctuate about average values and are characterized by probability distributions . While classical thermodynamics

3286-431: Is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium , and the microscopic behaviours and motions occurring inside the material. Whereas statistical mechanics proper involves dynamics, here

Statistical mechanics - Misplaced Pages Continue

3392-411: Is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes , a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within

3498-431: Is using physics or conducting physics research with the aim of developing new technologies or solving a problem. The approach is similar to that of applied mathematics . Applied physicists use physics in scientific research. For instance, people working on accelerator physics might seek to build better particle detectors for research in theoretical physics. Physics is used heavily in engineering. For example, statics,

3604-416: Is usual for probabilities, the ensemble can be interpreted in different ways: These two meanings are equivalent for many purposes, and will be used interchangeably in this article. However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in

3710-459: The Archaic period (650 BCE – 480 BCE), when pre-Socratic philosophers like Thales rejected non-naturalistic explanations for natural phenomena and proclaimed that every event had a natural cause. They proposed ideas verified by reason and observation, and many of their hypotheses proved successful in experiment; for example, atomism was found to be correct approximately 2000 years after it

3816-536: The Industrial Revolution as energy needs increased. The laws comprising classical physics remain widely used for objects on everyday scales travelling at non-relativistic speeds, since they provide a close approximation in such situations, and theories such as quantum mechanics and the theory of relativity simplify to their classical equivalents at such scales. Inaccuracies in classical mechanics for very small objects and very high velocities led to

3922-478: The Islamic Golden Age developed it further, especially placing emphasis on observation and a priori reasoning, developing early forms of the scientific method . The most notable innovations under Islamic scholarship were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl , Al-Kindi , Ibn al-Haytham , Al-Farisi and Avicenna . The most notable work

4028-590: The Scientific Revolution . Galileo cited Philoponus substantially in his works when arguing that Aristotelian physics was flawed. In the 1300s Jean Buridan , a teacher in the faculty of arts at the University of Paris , developed the concept of impetus. It was a step toward the modern ideas of inertia and momentum. Islamic scholarship inherited Aristotelian physics from the Greeks and during

4134-619: The Standard Model of particle physics was derived. Following the discovery of a particle with properties consistent with the Higgs boson at CERN in 2012, all fundamental particles predicted by the standard model, and no others, appear to exist; however, physics beyond the Standard Model , with theories such as supersymmetry , is an active area of research. Areas of mathematics in general are important to this field, such as

4240-579: The empirical world. This is usually combined with the claim that the laws of logic express universal regularities found in the structural features of the world, which may explain the peculiar relation between these fields. Physics uses mathematics to organise and formulate experimental results. From those results, precise or estimated solutions are obtained, or quantitative results, from which new predictions can be made and experimentally confirmed or negated. The results from physics experiments are numerical data, with their units of measure and estimates of

4346-598: The exact sciences are descended from late Babylonian astronomy . Egyptian astronomers left monuments showing knowledge of the constellations and the motions of the celestial bodies, while Greek poet Homer wrote of various celestial objects in his Iliad and Odyssey ; later Greek astronomers provided names, which are still used today, for most constellations visible from the Northern Hemisphere . Natural philosophy has its origins in Greece during

Statistical mechanics - Misplaced Pages Continue

4452-543: The standard consensus that the laws of physics are universal and do not change with time, physics can be used to study things that would ordinarily be mired in uncertainty . For example, in the study of the origin of the Earth, a physicist can reasonably model Earth's mass, temperature, and rate of rotation, as a function of time allowing the extrapolation forward or backward in time and so predict future or prior events. It also allows for simulations in engineering that speed up

4558-432: The von Neumann equation . These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. These ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy

4664-560: The "interesting" information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors ), where the electrons are indeed analogous to a rarefied gas. Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations,

4770-435: The 16th and 17th centuries, and Isaac Newton 's discovery and unification of the laws of motion and universal gravitation (that would come to bear his name). Newton also developed calculus , the mathematical study of continuous change, which provided new mathematical methods for solving physical problems. The discovery of laws in thermodynamics , chemistry , and electromagnetics resulted from research efforts during

4876-574: The area of medical diagnostics . Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems . In quantum mechanics, a statistical ensemble (probability distribution over possible quantum states ) is described by a density operator S , which is a non-negative, self-adjoint , trace-class operator of trace 1 on the Hilbert space H describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics . One such formalism

4982-623: The attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving ( mechanical equilibrium ), rather, only that the ensemble is not evolving. A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.). There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why

5088-446: The book was to distill these results into a cohesive and simple picture. Gibbs wrote in 1892 to his colleague Lord Rayleigh Just now I am trying to get ready for publication something on thermodynamics from the a-priori point of view, or rather on 'statistical mechanics' [...] I do not know that I shall have anything particularly new in substance, but shall be contented if I can so choose my standpoint (as seems to me possible) as to get

5194-433: The boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in these and other academic disciplines such as mathematics and philosophy. Advances in physics often enable new technologies . For example, advances in the understanding of electromagnetism , solid-state physics , and nuclear physics led directly to

5300-650: The characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities. There are some cases which allow exact solutions. Although some problems in statistical physics can be solved analytically using approximations and expansions, most current research utilizes

5406-434: The concepts of space, time, and matter from that presented by classical physics. Classical mechanics approximates nature as continuous, while quantum theory is concerned with the discrete nature of many phenomena at the atomic and subatomic level and with the complementary aspects of particles and waves in the description of such phenomena. The theory of relativity is concerned with the description of phenomena that take place in

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5512-629: The conductance of an electronic system is the use of the Green–Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method. The ensemble formalism can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in: Statistical physics explains and quantitatively describes superconductivity , superfluidity , turbulence , collective phenomena in solids and plasma , and

5618-409: The constant speed predicted by Maxwell's equations of electromagnetism. This discrepancy was corrected by Einstein's theory of special relativity , which replaced classical mechanics for fast-moving bodies and allowed for a constant speed of light. Black-body radiation provided another problem for classical physics, which was corrected when Planck proposed that the excitation of material oscillators

5724-415: The density object it is falling through (e.g. density of air). He also stated that, when it comes to violent motion (motion of an object when a force is applied to it by a second object) that the speed that object moves, will only be as fast or strong as the measure of force applied to it. The problem of motion and its causes was studied carefully, leading to the philosophical notion of a " prime mover " as

5830-466: The development of a new technology. There is also considerable interdisciplinarity , so many other important fields are influenced by physics (e.g., the fields of econophysics and sociophysics ). Physicists use the scientific method to test the validity of a physical theory . By using a methodical approach to compare the implications of a theory with the conclusions drawn from its related experiments and observations, physicists are better able to test

5936-429: The development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory and Albert Einstein 's theory of relativity. Both of these theories came about due to inaccuracies in classical mechanics in certain situations. Classical mechanics predicted that the speed of light depends on the motion of the observer, which could not be resolved with

6042-407: The development of new experiments (and often related equipment). Physicists who work at the interplay of theory and experiment are called phenomenologists , who study complex phenomena observed in experiment and work to relate them to a fundamental theory . Theoretical physics has historically taken inspiration from philosophy; electromagnetism was unified this way. Beyond the known universe,

6148-556: The development of technologies that have transformed modern society, such as television, computers, domestic appliances , and nuclear weapons ; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus . The word physics comes from the Latin physica ('study of nature'), which itself is a borrowing of the Greek φυσική ( phusikḗ 'natural science'),

6254-422: The difference in the weights is not considerable, that is, of one is, let us say, double the other, there will be no difference, or else an imperceptible difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as the other Philoponus' criticism of Aristotelian principles of physics served as an inspiration for Galileo Galilei ten centuries later, during

6360-536: The ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state. One special class of ensemble

6466-520: The ensemble for a given system should have one form or another. A common approach found in many textbooks is to take the equal a priori probability postulate . This postulate states that The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate: Other fundamental postulates for statistical mechanics have also been proposed. For example, recent studies shows that

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6572-575: The ensemble if the ‘ergodic hypothesis’ is valid. The ergodic hypothesis, which states that all the microstates of the system are sampled with equal probability, is applicable to most systems, with the exception of systems such as quenched glasses which are in metastable states. Thus, the ensemble averaging method provides us an easy way to calculate the thermodynamic properties of the system, without having to observe it for long periods of time. Gibbs also used this tool to obtain relationships between systems constrained in different ways, for example, to relate

6678-682: The errors in the measurements. Technologies based on mathematics, like computation have made computational physics an active area of research. Ontology is a prerequisite for physics, but not for mathematics. It means physics is ultimately concerned with descriptions of the real world, while mathematics is concerned with abstract patterns, even beyond the real world. Thus physics statements are synthetic, while mathematical statements are analytic. Mathematics contains hypotheses, while physics contains theories. Mathematics statements have to be only logically true, while predictions of physics statements must match observed and experimental data. The distinction

6784-476: The explanations for the observed positions of the stars were often unscientific and lacking in evidence, these early observations laid the foundation for later astronomy, as the stars were found to traverse great circles across the sky, which could not explain the positions of the planets . According to Asger Aaboe , the origins of Western astronomy can be found in Mesopotamia , and all Western efforts in

6890-996: The field of theoretical physics also deals with hypothetical issues, such as parallel universes , a multiverse , and higher dimensions . Theorists invoke these ideas in hopes of solving particular problems with existing theories; they then explore the consequences of these ideas and work toward making testable predictions. Experimental physics expands, and is expanded by, engineering and technology. Experimental physicists who are involved in basic research design and perform experiments with equipment such as particle accelerators and lasers , whereas those involved in applied research often work in industry, developing technologies such as magnetic resonance imaging (MRI) and transistors . Feynman has noted that experimentalists may seek areas that have not been explored well by theorists. Elementary Principles in Statistical Mechanics In this book, Gibbs carefully showed how

6996-410: The fluctuation–dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics. A few of the theoretical tools used to make this connection include: An advanced approach uses a combination of stochastic methods and linear response theory . As an example, one approach to compute quantum coherence effects ( weak localization , conductance fluctuations ) in

7102-445: The knowledge of previous scholars, he began to explain how light enters the eye. He asserted that the light ray is focused, but the actual explanation of how light projected to the back of the eye had to wait until 1604. His Treatise on Light explained the camera obscura , hundreds of years before the modern development of photography. The seven-volume Book of Optics ( Kitab al-Manathir ) influenced thinking across disciplines from

7208-515: The large processing power of modern computers to simulate or approximate solutions. A common approach to statistical problems is to use a Monte Carlo simulation to yield insight into the properties of a complex system . Monte Carlo methods are important in computational physics , physical chemistry , and related fields, and have diverse applications including medical physics , where they are used to model radiation transport for radiation dosimetry calculations. The Monte Carlo method examines just

7314-408: The later quantum mechanics , and still form the foundation of statistical mechanics to this day. In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics . For both types of mechanics, the standard mathematical approach is to consider two concepts: Using these two concepts, the state at any other time, past or future, can in principle be calculated. There

7420-400: The latter include such branches as hydrostatics , hydrodynamics and pneumatics . Acoustics is the study of how sound is produced, controlled, transmitted and received. Important modern branches of acoustics include ultrasonics , the study of sound waves of very high frequency beyond the range of human hearing; bioacoustics , the physics of animal calls and hearing, and electroacoustics ,

7526-413: The laws of thermodynamics would arise exactly from a generic classical mechanical system, if one allowed for a certain natural uncertainty about the state of that system. The themes of thermodynamic connections to statistical mechanics had been explored in the preceding decades with Clausius , Maxwell , and Boltzmann , together writing thousands of pages on this topic. One of Gibbs' aims in writing

7632-490: The laws of classical physics accurately describe systems whose important length scales are greater than the atomic scale and whose motions are much slower than the speed of light. Outside of this domain, observations do not match predictions provided by classical mechanics. Einstein contributed the framework of special relativity, which replaced notions of absolute time and space with spacetime and allowed an accurate description of systems whose components have speeds approaching

7738-409: The major upheavals of modern physics during the early 20th century. V. Kumaran wrote the following comment regarding Elementary Principles in Statistical Mechanics : ... In this, he introduced the now standard concept of ‘ensemble’, which is a collection of a large number of indistinguishable replicas of the system under consideration, which interact with each other, but which are isolated from

7844-412: The manipulation of audible sound waves using electronics. Optics, the study of light, is concerned not only with visible light but also with infrared and ultraviolet radiation , which exhibit all of the phenomena of visible light except visibility, e.g., reflection, refraction, interference, diffraction, dispersion, and polarization of light. Heat is a form of energy, the internal energy possessed by

7950-429: The natural place of another, the less abundant element will automatically go towards its own natural place. For example, if there is a fire on the ground, the flames go up into the air in an attempt to go back into its natural place where it belongs. His laws of motion included: that heavier objects will fall faster, the speed being proportional to the weight and the speed of the object that is falling depends inversely on

8056-423: The oldest academic disciplines . Over much of the past two millennia, physics, chemistry , biology , and certain branches of mathematics were a part of natural philosophy , but during the Scientific Revolution in the 17th century, these natural sciences branched into separate research endeavors. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry , and

8162-572: The particles of which a substance is composed; thermodynamics deals with the relationships between heat and other forms of energy. Electricity and magnetism have been studied as a single branch of physics since the intimate connection between them was discovered in the early 19th century; an electric current gives rise to a magnetic field , and a changing magnetic field induces an electric current. Electrostatics deals with electric charges at rest, electrodynamics with moving charges, and magnetostatics with magnetic poles at rest. Classical physics

8268-427: The practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in. Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble , which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of

8374-569: The proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H -theorem . The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in 1884. According to Gibbs, the term "statistical", in the context of mechanics, i.e. statistical mechanics,

8480-474: The proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. Five years later, in 1864, Ludwig Boltzmann , a young student in Vienna, came across Maxwell's paper and spent much of his life developing

8586-432: The response can be analysed in linear response theory . A remarkable result, as formalized by the fluctuation–dissipation theorem , is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in

8692-407: The rest of the universe. The replicas could be in different microscopic states, as determined by the positions and momenta of the constituent molecules, for example, but the macroscopic state determined by the pressure, temperature and / or other thermodynamic variables are identical. Gibbs argued that the properties of the system, averaged over time, is identical to an average over all the members of

8798-406: The same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium. This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations,

8904-412: The simplest non-equilibrium situation of a steady state current flow in a system of many particles. In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases . In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on

9010-599: The sixth century, Isidore of Miletus created an important compilation of Archimedes ' works that are copied in the Archimedes Palimpsest . In sixth-century Europe John Philoponus , a Byzantine scholar, questioned Aristotle 's teaching of physics and noted its flaws. He introduced the theory of impetus . Aristotle's physics was not scrutinized until Philoponus appeared; unlike Aristotle, who based his physics on verbal argument, Philoponus relied on observation. On Aristotle's physics Philoponus wrote: But this

9116-444: The size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system. Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from

9222-412: The speed of light. Planck, Schrödinger, and others introduced quantum mechanics, a probabilistic notion of particles and interactions that allowed an accurate description of atomic and subatomic scales. Later, quantum field theory unified quantum mechanics and special relativity. General relativity allowed for a dynamical, curved spacetime, with which highly massive systems and the large-scale structure of

9328-419: The structural features of liquid . It underlies the modern astrophysics . In solid state physics, statistical physics aids the study of liquid crystals , phase transitions , and critical phenomena . Many experimental studies of matter are entirely based on the statistical description of a system. These include the scattering of cold neutrons , X-ray , visible light , and more. Statistical physics also plays

9434-412: The study of probabilities and groups . Physics deals with a wide variety of systems, although certain theories are used by all physicists. Each of these theories was experimentally tested numerous times and found to be an adequate approximation of nature. For instance, the theory of classical mechanics accurately describes the motion of objects, provided they are much larger than atoms and moving at

9540-459: The subject further. Statistical mechanics was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory . Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem , transport theory , thermal equilibrium , the equation of state of gases, and similar subjects, occupy about 2,000 pages in

9646-462: The system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier. The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where

9752-413: The system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinate axes. In quantum statistical mechanics, the ensemble is a probability distribution over pure states and can be compactly summarized as a density matrix . As

9858-425: The theory of four elements . Aristotle believed that each of the four classical elements (air, fire, water, earth) had its own natural place. Because of their differing densities, each element will revert to its own specific place in the atmosphere. So, because of their weights, fire would be at the top, air underneath fire, then water, then lastly earth. He also stated that when a small amount of one element enters

9964-399: The theory of statistical mechanics can be built without the equal a priori probability postulate. One such formalism is based on the fundamental thermodynamic relation together with the following set of postulates: where the third postulate can be replaced by the following: There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside

10070-561: The theory of visual perception to the nature of perspective in medieval art, in both the East and the West, for more than 600 years. This included later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to Johannes Kepler . The translation of The Book of Optics had an impact on Europe. From it, later European scholars were able to build devices that replicated those Ibn al-Haytham had built and understand

10176-520: The time of the book's writing, the prevailing understanding of nature was purely in classical terms: Quantum mechanics had not yet been conceived, and even basic facts taken for granted today (such as the existence of atoms) were still contested among scientists. Gibbs was careful in assuming the least about the nature of physical systems under study, and as a result the principles of statistical mechanics laid down by Gibbs have retained their accuracy (with some changes in detail but not in theme), in spite of

10282-569: The ultimate source of all motion in the world (Book 8 of his treatise Physics ). The Western Roman Empire fell to invaders and internal decay in the fifth century, resulting in a decline in intellectual pursuits in western Europe. By contrast, the Eastern Roman Empire (usually known as the Byzantine Empire ) resisted the attacks from invaders and continued to advance various fields of learning, including physics. In

10388-423: The universe can be well-described. General relativity has not yet been unified with the other fundamental descriptions; several candidate theories of quantum gravity are being developed. Physics, as with the rest of science, relies on the philosophy of science and its " scientific method " to advance knowledge of the physical world. The scientific method employs a priori and a posteriori reasoning as well as

10494-573: The use of Bayesian inference to measure the validity of a given theory. Study of the philosophical issues surrounding physics, the philosophy of physics , involves issues such as the nature of space and time , determinism , and metaphysical outlooks such as empiricism , naturalism , and realism . Many physicists have written about the philosophical implications of their work, for instance Laplace , who championed causal determinism , and Erwin Schrödinger , who wrote on quantum mechanics. The mathematical physicist Roger Penrose has been called

10600-988: The validity of a theory in a logical, unbiased, and repeatable way. To that end, experiments are performed and observations are made in order to determine the validity or invalidity of a theory. A scientific law is a concise verbal or mathematical statement of a relation that expresses a fundamental principle of some theory, such as Newton's law of universal gravitation. Theorists seek to develop mathematical models that both agree with existing experiments and successfully predict future experimental results, while experimentalists devise and perform experiments to test theoretical predictions and explore new phenomena. Although theory and experiment are developed separately, they strongly affect and depend upon each other. Progress in physics frequently comes about when experimental results defy explanation by existing theories, prompting intense focus on applicable modelling, and when new theories generate experimentally testable predictions , which inspire

10706-579: The way vision works. Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics . Major developments in this period include the replacement of the geocentric model of the Solar System with the heliocentric Copernican model , the laws governing the motion of planetary bodies (determined by Kepler between 1609 and 1619), Galileo's pioneering work on telescopes and observational astronomy in

10812-410: Was The Book of Optics (also known as Kitāb al-Manāẓir), written by Ibn al-Haytham, in which he presented the alternative to the ancient Greek idea about vision. In his Treatise on Light as well as in his Kitāb al-Manāẓir , he presented a study of the phenomenon of the camera obscura (his thousand-year-old version of the pinhole camera ) and delved further into the way the eye itself works. Using

10918-444: Was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology. Important cases where the thermodynamic ensembles do not give identical results include: In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in

11024-496: Was first used by the Scottish physicist James Clerk Maxwell in 1871: "In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus." "Probabilistic mechanics" might today seem a more appropriate term, but "statistical mechanics"

11130-487: Was influential for about two millennia. His approach mixed some limited observation with logical deductive arguments, but did not rely on experimental verification of deduced statements. Aristotle's foundational work in Physics, though very imperfect, formed a framework against which later thinkers further developed the field. His approach is entirely superseded today. He explained ideas such as motion (and gravity ) with

11236-546: Was proposed by Leucippus and his pupil Democritus . During the classical period in Greece (6th, 5th and 4th centuries BCE) and in Hellenistic times , natural philosophy developed along many lines of inquiry. Aristotle ( Greek : Ἀριστοτέλης , Aristotélēs ) (384–322 BCE), a student of Plato , wrote on many subjects, including a substantial treatise on " Physics " – in the 4th century BC. Aristotelian physics

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