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97-546: The significance of the virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution, such as those considered in statistical mechanics ; this average total kinetic energy is related to the temperature of the system by the equipartition theorem . However, the virial theorem does not depend on the notion of temperature and holds even for systems that are not in thermal equilibrium . The virial theorem has been generalized in various ways, most notably to

194-441: A tensor form. If the force between any two particles of the system results from a potential energy V ( r ) = αr that is proportional to some power n of the interparticle distance r , the virial theorem takes the simple form 2 ⟨ T ⟩ = n ⟨ V TOT ⟩ . {\displaystyle 2\langle T\rangle =n\langle V_{\text{TOT}}\rangle .} Thus, twice

291-1752: A common special case, the potential energy V between two particles is proportional to a power n of their distance r ij : V j k = α r j k n , {\displaystyle V_{jk}=\alpha r_{jk}^{n},} where the coefficient α and the exponent n are constants. In such cases, the virial is − 1 2 ∑ k = 1 N F k ⋅ r k = 1 2 ∑ k = 1 N ∑ j < k d V j k d r j k r j k = 1 2 ∑ k = 1 N ∑ j < k n α r j k n − 1 r j k = 1 2 ∑ k = 1 N ∑ j < k n V j k = n 2 V TOT , {\displaystyle {\begin{aligned}-{\frac {1}{2}}\,\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}&={\frac {1}{2}}\,\sum _{k=1}^{N}\sum _{j<k}{\frac {dV_{jk}}{dr_{jk}}}r_{jk}\\&={\frac {1}{2}}\,\sum _{k=1}^{N}\sum _{j<k}n\alpha r_{jk}^{n-1}r_{jk}\\&={\frac {1}{2}}\,\sum _{k=1}^{N}\sum _{j<k}nV_{jk}={\frac {n}{2}}\,V_{\text{TOT}},\end{aligned}}} where V TOT = ∑ k = 1 N ∑ j < k V j k {\displaystyle V_{\text{TOT}}=\sum _{k=1}^{N}\sum _{j<k}V_{jk}}

388-445: A container filled with an ideal gas consisting of point masses. The force applied to the point masses is the negative of the forces applied to the wall of the container, which is of the form d F = − n ^ P d A {\displaystyle d\mathbf {F} =-{\hat {\mathbf {n} }}PdA} , where n ^ {\displaystyle {\hat {\mathbf {n} }}}

485-692: A duration τ is defined as ⟨ d G d t ⟩ τ = 1 τ ∫ 0 τ d G d t d t = 1 τ ∫ G ( 0 ) G ( τ ) d G = G ( τ ) − G ( 0 ) τ , {\displaystyle \left\langle {\frac {dG}{dt}}\right\rangle _{\tau }={\frac {1}{\tau }}\int _{0}^{\tau }{\frac {dG}{dt}}\,dt={\frac {1}{\tau }}\int _{G(0)}^{G(\tau )}\,dG={\frac {G(\tau )-G(0)}{\tau }},} from which we obtain

582-601: 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

679-455: 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

776-412: A one-dimensional oscillator with mass m {\displaystyle m} , position x {\displaystyle x} , driving force F cos ⁡ ( ω t ) {\displaystyle F\cos(\omega t)} , spring constant k {\displaystyle k} , and damping coefficient γ {\displaystyle \gamma } ,

873-424: 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

970-1593: A single particle in special relativity, it is not the case that T = ⁠ 1 / 2 ⁠ p · v . Instead, it is true that T = ( γ − 1) mc , where γ is the Lorentz factor γ = 1 1 − v 2 c 2 , {\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},} and β = ⁠ v / c ⁠ . We have 1 2 p ⋅ v = 1 2 β γ m c ⋅ β c = 1 2 γ β 2 m c 2 = ( γ β 2 2 ( γ − 1 ) ) T . {\displaystyle {\begin{aligned}{\frac {1}{2}}\mathbf {p} \cdot \mathbf {v} &={\frac {1}{2}}{\boldsymbol {\beta }}\gamma mc\cdot {\boldsymbol {\beta }}c\\&={\frac {1}{2}}\gamma \beta ^{2}mc^{2}\\[5pt]&=\left({\frac {\gamma \beta ^{2}}{2(\gamma -1)}}\right)T.\end{aligned}}} The last expression can be simplified to ( 1 + 1 − β 2 2 ) T = ( γ + 1 2 γ ) T . {\displaystyle \left({\frac {1+{\sqrt {1-\beta ^{2}}}}{2}}\right)T=\left({\frac {\gamma +1}{2\gamma }}\right)T.} Thus, under

1067-730: A solution to the equation − ∇ 2 u = g ( u ) , {\displaystyle -\nabla ^{2}u=g(u),} in the sense of distributions . Then u {\displaystyle u} satisfies the relation ( n − 2 2 ) ∫ R n | ∇ u ( x ) | 2 d x = n ∫ R n G ( u ( x ) ) d x . {\displaystyle \left({\frac {n-2}{2}}\right)\int _{\mathbb {R} ^{n}}|\nabla u(x)|^{2}\,dx=n\int _{\mathbb {R} ^{n}}G{\big (}u(x){\big )}\,dx.} For

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1164-407: 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)

1261-482: 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

1358-521: A system is a star held together by its own gravity, where n = −1 . In 1870, Rudolf Clausius delivered the lecture "On a Mechanical Theorem Applicable to Heat" to the Association for Natural and Medical Sciences of the Lower Rhine, following a 20-year study of thermodynamics. The lecture stated that the mean vis viva of the system is equal to its virial, or that the average kinetic energy

1455-671: Is 1 2 d 2 I d t 2 + ∫ V x k ∂ G k ∂ t d 3 r = 2 ( T + U ) + W E + W M − ∫ x k ( p i k + T i k ) d S i , {\displaystyle {\frac {1}{2}}{\frac {d^{2}I}{dt^{2}}}+\int _{V}x_{k}{\frac {\partial G_{k}}{\partial t}}\,d^{3}r=2(T+U)+W^{\mathrm {E} }+W^{\mathrm {M} }-\int x_{k}(p_{ik}+T_{ik})\,dS_{i},} where I

1552-404: 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 is to clarify

1649-499: 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,

1746-2126: Is equal and opposite to F kj = −∇ r j V kj = −∇ r j V jk , the force applied by particle k on particle j , as may be confirmed by explicit calculation. Hence, ∑ k = 1 N F k ⋅ r k = ∑ k = 2 N ∑ j = 1 k − 1 F j k ⋅ ( r k − r j ) = − ∑ k = 2 N ∑ j = 1 k − 1 d V j k d r j k | r k − r j | 2 r j k = − ∑ k = 2 N ∑ j = 1 k − 1 d V j k d r j k r j k . {\displaystyle {\begin{aligned}\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}&=\sum _{k=2}^{N}\sum _{j=1}^{k-1}\mathbf {F} _{jk}\cdot (\mathbf {r} _{k}-\mathbf {r} _{j})\\&=-\sum _{k=2}^{N}\sum _{j=1}^{k-1}{\frac {dV_{jk}}{dr_{jk}}}{\frac {|\mathbf {r} _{k}-\mathbf {r} _{j}|^{2}}{r_{jk}}}\\&=-\sum _{k=2}^{N}\sum _{j=1}^{k-1}{\frac {dV_{jk}}{dr_{jk}}}r_{jk}.\end{aligned}}} Thus d G d t = 2 T + ∑ k = 1 N F k ⋅ r k = 2 T − ∑ k = 2 N ∑ j = 1 k − 1 d V j k d r j k r j k . {\displaystyle {\frac {dG}{dt}}=2T+\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}=2T-\sum _{k=2}^{N}\sum _{j=1}^{k-1}{\frac {dV_{jk}}{dr_{jk}}}r_{jk}.} In

1843-513: 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

1940-417: 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

2037-516: Is one half of the average potential energy. The virial theorem can be obtained directly from Lagrange's identity as applied in classical gravitational dynamics, the original form of which was included in Lagrange's "Essay on the Problem of Three Bodies" published in 1772. Carl Jacobi's generalization of the identity to N  bodies and to the present form of Laplace's identity closely resembles

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2134-449: 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

2231-469: 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

2328-891: Is the moment of inertia , G is the momentum density of the electromagnetic field , T is the kinetic energy of the "fluid", U is the random "thermal" energy of the particles, W and W are the electric and magnetic energy content of the volume considered. Finally, p ik is the fluid-pressure tensor expressed in the local moving coordinate system p i k = Σ n σ m σ ⟨ v i v k ⟩ σ − V i V k Σ m σ n σ , {\displaystyle p_{ik}=\Sigma n^{\sigma }m^{\sigma }\langle v_{i}v_{k}\rangle ^{\sigma }-V_{i}V_{k}\Sigma m^{\sigma }n^{\sigma },} Statistical mechanics In physics , statistical mechanics

2425-633: Is the kinetic energy. The left-hand side of this equation is just dQ / dt , according to the Heisenberg equation of motion. The expectation value ⟨ dQ / dt ⟩ of this time derivative vanishes in a stationary state, leading to the quantum virial theorem : 2 ⟨ T ⟩ = ∑ n ⟨ X n d V d X n ⟩ . {\displaystyle 2\langle T\rangle =\sum _{n}\left\langle X_{n}{\frac {dV}{dX_{n}}}\right\rangle .} In

2522-844: Is the mass of the k th particle, F k = ⁠ d p k / dt ⁠ is the net force on that particle, and T is the total kinetic energy of the system according to the v k = ⁠ d r k / dt ⁠ velocity of each particle, T = 1 2 ∑ k = 1 N m k v k 2 = 1 2 ∑ k = 1 N m k d r k d t ⋅ d r k d t . {\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}v_{k}^{2}={\frac {1}{2}}\sum _{k=1}^{N}m_{k}{\frac {d\mathbf {r} _{k}}{dt}}\cdot {\frac {d\mathbf {r} _{k}}{dt}}.} The total force F k on particle k

2619-940: Is the natural frequency of the oscillator. To solve the two unknowns, we need another equation. In steady state, the power lost per cycle is equal to the power gained per cycle: ⟨ x ˙ γ x ˙ ⟩ ⏟ power dissipated = ⟨ x ˙ F cos ⁡ ω t ⟩ ⏟ power input , {\displaystyle \underbrace {\langle {\dot {x}}\,\gamma {\dot {x}}\rangle } _{\text{power dissipated}}=\underbrace {\langle {\dot {x}}\,F\cos \omega t\rangle } _{\text{power input}},} which simplifies to sin ⁡ φ = − γ X ω F {\displaystyle \sin \varphi =-{\frac {\gamma X\omega }{F}}} . Now we have two equations that yield

2716-993: Is the sum of all the forces from the other particles j in the system: F k = ∑ j = 1 N F j k , {\displaystyle \mathbf {F} _{k}=\sum _{j=1}^{N}\mathbf {F} _{jk},} where F jk is the force applied by particle j on particle k . Hence, the virial can be written as − 1 2 ∑ k = 1 N F k ⋅ r k = − 1 2 ∑ k = 1 N ∑ j = 1 N F j k ⋅ r k . {\displaystyle -{\frac {1}{2}}\,\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}=-{\frac {1}{2}}\,\sum _{k=1}^{N}\sum _{j=1}^{N}\mathbf {F} _{jk}\cdot \mathbf {r} _{k}.} Since no particle acts on itself (i.e., F jj = 0 for 1 ≤ j ≤ N ), we split

2813-425: Is the total potential energy of the system. Thus d G d t = 2 T + ∑ k = 1 N F k ⋅ r k = 2 T − n V TOT . {\displaystyle {\frac {dG}{dt}}=2T+\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}=2T-nV_{\text{TOT}}.} For gravitating systems

2910-595: Is the unit normal vector pointing outwards. Then the virial theorem states that ⟨ T ⟩ = − 1 2 ⟨ ∑ i F i ⋅ r i ⟩ = P 2 ∫ n ^ ⋅ r d A . {\displaystyle \langle T\rangle =-{\frac {1}{2}}{\Big \langle }\sum _{i}\mathbf {F} _{i}\cdot \mathbf {r} _{i}{\Big \rangle }={\frac {P}{2}}\int {\hat {\mathbf {n} }}\cdot \mathbf {r} \,dA.} By

3007-400: 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

Virial theorem - Misplaced Pages Continue

3104-431: 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

3201-411: 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

3298-417: 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

3395-816: The divergence theorem , ∫ n ^ ⋅ r d A = ∫ ∇ ⋅ r d V = 3 ∫ d V = 3 V {\textstyle \int {\hat {\mathbf {n} }}\cdot \mathbf {r} \,dA=\int \nabla \cdot \mathbf {r} \,dV=3\int dV=3V} . And since the average total kinetic energy ⟨ T ⟩ = N ⟨ 1 2 m v 2 ⟩ = N ⋅ 3 2 k T {\textstyle \langle T\rangle =N{\big \langle }{\frac {1}{2}}mv^{2}{\big \rangle }=N\cdot {\frac {3}{2}}kT} , we have P V = N k T {\displaystyle PV=NkT} . In 1933, Fritz Zwicky applied

3492-1025: The k th particle. Assuming that the masses are constant, G is one-half the time derivative of this moment of inertia: 1 2 d I d t = 1 2 d d t ∑ k = 1 N m k r k ⋅ r k = ∑ k = 1 N m k d r k d t ⋅ r k = ∑ k = 1 N p k ⋅ r k = G . {\displaystyle {\begin{aligned}{\frac {1}{2}}{\frac {dI}{dt}}&={\frac {1}{2}}{\frac {d}{dt}}\sum _{k=1}^{N}m_{k}\mathbf {r} _{k}\cdot \mathbf {r} _{k}\\&=\sum _{k=1}^{N}m_{k}\,{\frac {d\mathbf {r} _{k}}{dt}}\cdot \mathbf {r} _{k}\\&=\sum _{k=1}^{N}\mathbf {p} _{k}\cdot \mathbf {r} _{k}=G.\end{aligned}}} In turn,

3589-444: The scalar moment of inertia I about the origin is I = ∑ k = 1 N m k | r k | 2 = ∑ k = 1 N m k r k 2 , {\displaystyle I=\sum _{k=1}^{N}m_{k}|\mathbf {r} _{k}|^{2}=\sum _{k=1}^{N}m_{k}r_{k}^{2},} where m k and r k represent

3686-432: 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

3783-560: 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,

3880-453: 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

3977-625: 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

Virial theorem - Misplaced Pages Continue

4074-492: The average kinetic energy equals half of the average negative potential energy: ⟨ T ⟩ τ = − 1 2 ⟨ V TOT ⟩ τ . {\displaystyle \langle T\rangle _{\tau }=-{\frac {1}{2}}\langle V_{\text{TOT}}\rangle _{\tau }.} This general result is useful for complex gravitating systems such as planetary systems or galaxies . A simple application of

4171-1195: The average of the time derivative might vanish. One often-cited reason applies to stably bound systems, that is, to systems that hang together forever and whose parameters are finite. In this case, velocities and coordinates of the particles of the system have upper and lower limits, so that G is bounded between two extremes, G min and G max , and the average goes to zero in the limit of infinite τ : lim τ → ∞ | ⟨ d G bound d t ⟩ τ | = lim τ → ∞ | G ( τ ) − G ( 0 ) τ | ≤ lim τ → ∞ G max − G min τ = 0. {\displaystyle \lim _{\tau \to \infty }\left|\left\langle {\frac {dG^{\text{bound}}}{dt}}\right\rangle _{\tau }\right|=\lim _{\tau \to \infty }\left|{\frac {G(\tau )-G(0)}{\tau }}\right|\leq \lim _{\tau \to \infty }{\frac {G_{\max }-G_{\min }}{\tau }}=0.} Even if

4268-797: The average of the time derivative of G is only approximately zero, the virial theorem holds to the same degree of approximation. For power-law forces with an exponent n , the general equation holds: ⟨ T ⟩ τ = − 1 2 ∑ k = 1 N ⟨ F k ⋅ r k ⟩ τ = n 2 ⟨ V TOT ⟩ τ . {\displaystyle \langle T\rangle _{\tau }=-{\frac {1}{2}}\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle _{\tau }={\frac {n}{2}}\langle V_{\text{TOT}}\rangle _{\tau }.} For gravitational attraction, n = −1 , and

4365-411: The average total kinetic energy ⟨ T ⟩ equals n times the average total potential energy ⟨ V TOT ⟩ . Whereas V ( r ) represents the potential energy between two particles of distance r , V TOT represents the total potential energy of the system, i.e., the sum of the potential energy V ( r ) over all pairs of particles in the system. A common example of such

4462-650: 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

4559-513: The classical virial theorem. However, the interpretations leading to the development of the equations were very different, since at the time of development, statistical dynamics had not yet unified the separate studies of thermodynamics and classical dynamics. The theorem was later utilized, popularized, generalized and further developed by James Clerk Maxwell , Lord Rayleigh , Henri Poincaré , Subrahmanyan Chandrasekhar , Enrico Fermi , Paul Ledoux , Richard Bader and Eugene Parker . Fritz Zwicky

4656-531: The cluster is U = − ∑ i < j G m 2 r i , j {\displaystyle U=-\sum _{i<j}{\frac {Gm^{2}}{r_{i,j}}}} , giving ⟨ U ⟩ = − G m 2 ∑ i < j ⟨ 1 / r i , j ⟩ {\textstyle \langle U\rangle =-Gm^{2}\sum _{i<j}\langle {1}/{r_{i,j}}\rangle } . Assuming

4753-843: The cluster, each having observed stellar mass m = 10 9 M ⊙ {\displaystyle m=10^{9}M_{\odot }} (suggested by Hubble), and the cluster has radius R = 10 6 ly {\displaystyle R=10^{6}{\text{ly}}} . He also measured the radial velocities of the galaxies by doppler shifts in galactic spectra to be ⟨ v r 2 ⟩ = ( 1000 km/s ) 2 {\displaystyle \langle v_{r}^{2}\rangle =(1000{\text{km/s}})^{2}} . Assuming equipartition of kinetic energy, ⟨ v 2 ⟩ = 3 ⟨ v r 2 ⟩ {\displaystyle \langle v^{2}\rangle =3\langle v_{r}^{2}\rangle } . By

4850-453: The commutator is i ℏ [ H , Q ] = 2 T − ∑ n X n d V d X n , {\displaystyle {\frac {i}{\hbar }}[H,Q]=2T-\sum _{n}X_{n}{\frac {dV}{dX_{n}}},} where T = ∑ n P n 2 / 2 m n {\textstyle T=\sum _{n}P_{n}^{2}/2m_{n}}

4947-1075: The conditions described in earlier sections (including Newton's third law of motion , F jk = − F kj , despite relativity), the time average for N particles with a power law potential is n 2 ⟨ V TOT ⟩ τ = ⟨ ∑ k = 1 N ( 1 + 1 − β k 2 2 ) T k ⟩ τ = ⟨ ∑ k = 1 N ( γ k + 1 2 γ k ) T k ⟩ τ . {\displaystyle {\frac {n}{2}}\left\langle V_{\text{TOT}}\right\rangle _{\tau }=\left\langle \sum _{k=1}^{N}\left({\tfrac {1+{\sqrt {1-\beta _{k}^{2}}}}{2}}\right)T_{k}\right\rangle _{\tau }=\left\langle \sum _{k=1}^{N}\left({\frac {\gamma _{k}+1}{2\gamma _{k}}}\right)T_{k}\right\rangle _{\tau }.} In particular,

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5044-631: 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

5141-536: 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

5238-524: 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

5335-844: The equation of motion is m d 2 x d t 2 ⏟ acceleration = − k x d d ⏟ spring   −   γ d x d t ⏟ friction   +   F cos ⁡ ( ω t ) d d ⏟ external driving . {\displaystyle m\underbrace {\frac {d^{2}x}{dt^{2}}} _{\text{acceleration}}=\underbrace {-kx{\vphantom {\frac {d}{d}}}} _{\text{spring}}\ \underbrace {-\ \gamma {\frac {dx}{dt}}} _{\text{friction}}\ \underbrace {+\ F\cos(\omega t){\vphantom {\frac {d}{d}}}} _{\text{external driving}}.} When

5432-1007: The exact equation ⟨ d G d t ⟩ τ = 2 ⟨ T ⟩ τ + ∑ k = 1 N ⟨ F k ⋅ r k ⟩ τ . {\displaystyle \left\langle {\frac {dG}{dt}}\right\rangle _{\tau }=2\langle T\rangle _{\tau }+\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle _{\tau }.} The virial theorem states that if ⟨ dG / dt ⟩ τ = 0 , then 2 ⟨ T ⟩ τ = − ∑ k = 1 N ⟨ F k ⋅ r k ⟩ τ . {\displaystyle 2\langle T\rangle _{\tau }=-\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle _{\tau }.} There are many reasons why

5529-449: The exponent n equals −1, giving Lagrange's identity d G d t = 1 2 d 2 I d t 2 = 2 T + V TOT , {\displaystyle {\frac {dG}{dt}}={\frac {1}{2}}{\frac {d^{2}I}{dt^{2}}}=2T+V_{\text{TOT}},} which was derived by Joseph-Louis Lagrange and extended by Carl Jacobi . The average of this derivative over

5626-1330: The field of quantum mechanics, there exists another form of the virial theorem, applicable to localized solutions to the stationary nonlinear Schrödinger equation or Klein–Gordon equation , is Pokhozhaev's identity , also known as Derrick's theorem . Let g ( s ) {\displaystyle g(s)} be continuous and real-valued, with g ( 0 ) = 0 {\displaystyle g(0)=0} . Denote G ( s ) = ∫ 0 s g ( t ) d t {\textstyle G(s)=\int _{0}^{s}g(t)\,dt} . Let u ∈ L loc ∞ ( R n ) , ∇ u ∈ L 2 ( R n ) , G ( u ( ⋅ ) ) ∈ L 1 ( R n ) , n ∈ N {\displaystyle u\in L_{\text{loc}}^{\infty }(\mathbb {R} ^{n}),\quad \nabla u\in L^{2}(\mathbb {R} ^{n}),\quad G(u(\cdot ))\in L^{1}(\mathbb {R} ^{n}),\quad n\in \mathbb {N} } be

5723-410: 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

5820-746: The forces can be derived from a potential energy V jk that is a function only of the distance r jk between the point particles j and k . Since the force is the negative gradient of the potential energy, we have in this case F j k = − ∇ r k V j k = − d V j k d r j k ( r k − r j r j k ) , {\displaystyle \mathbf {F} _{jk}=-\nabla _{\mathbf {r} _{k}}V_{jk}=-{\frac {dV_{jk}}{dr_{jk}}}\left({\frac {\mathbf {r} _{k}-\mathbf {r} _{j}}{r_{jk}}}\right),} which

5917-770: The gravitational potential of a uniform ball of constant density, giving ⟨ U ⟩ = − 3 5 G N 2 m 2 R {\textstyle \langle U\rangle =-{\frac {3}{5}}{\frac {GN^{2}m^{2}}{R}}} . So by the virial theorem, the total mass of the cluster is N m = 5 ⟨ v 2 ⟩ 3 G ⟨ 1 r ⟩ {\displaystyle Nm={\frac {5\langle v^{2}\rangle }{3G\langle {\frac {1}{r}}\rangle }}} Zwicky 1933 {\displaystyle _{1933}} estimated that there are N = 800 {\displaystyle N=800} galaxies in

SECTION 60

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6014-518: 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

6111-877: The larger ratios. The virial theorem has a particularly simple form for periodic motion. It can be used to perform perturbative calculation for nonlinear oscillators. It can also be used to study motion in a central potential . If the central potential is of the form U ∝ r n {\displaystyle U\propto r^{n}} , the virial theorem simplifies to ⟨ T ⟩ = n 2 ⟨ U ⟩ {\displaystyle \langle T\rangle ={\frac {n}{2}}\langle U\rangle } . In particular, for gravitational or electrostatic ( Coulomb ) attraction, ⟨ T ⟩ = − 1 2 ⟨ U ⟩ {\displaystyle \langle T\rangle =-{\frac {1}{2}}\langle U\rangle } . Analysis based on Sivardiere, 1986. For

6208-409: 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

6305-399: The mass and position of the k th particle. r k = | r k | is the position vector magnitude. Consider the scalar G = ∑ k = 1 N p k ⋅ r k , {\displaystyle G=\sum _{k=1}^{N}\mathbf {p} _{k}\cdot \mathbf {r} _{k},} where p k is the momentum vector of

6402-456: The motion of the stars are all the same over a long enough time ( ergodicity ), ⟨ U ⟩ = − 1 2 N 2 G m 2 ⟨ 1 / r ⟩ {\textstyle \langle U\rangle =-{\frac {1}{2}}N^{2}Gm^{2}\langle {1}/{r}\rangle } . Zwicky estimated ⟨ U ⟩ {\displaystyle \langle U\rangle } as

6499-403: The origin is displaced, then we'd obtain the same result. This is because the dot product of the displacement with equal and opposite forces F 1 ( t ) , F 2 ( t ) results in net cancellation. Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step. For a collection of N point particles,

6596-950: The origin, the particles have positions r 1 ( t ) and r 2 ( t ) = − r 1 ( t ) with fixed magnitude r . The attractive forces act in opposite directions as positions, so F 1 ( t ) ⋅ r 1 ( t ) = F 2 ( t ) ⋅ r 2 ( t ) = − Fr . Applying the centripetal force formula F = mv / r results in − 1 2 ∑ k = 1 N ⟨ F k ⋅ r k ⟩ = − 1 2 ( − F r − F r ) = F r = m v 2 r ⋅ r = m v 2 = ⟨ T ⟩ , {\displaystyle -{\frac {1}{2}}\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle =-{\frac {1}{2}}(-Fr-Fr)=Fr={\frac {mv^{2}}{r}}\cdot r=mv^{2}=\langle T\rangle ,} as required. Note: If

6693-1272: The oscillator has reached a steady state, it performs a stable oscillation x = X cos ⁡ ( ω t + φ ) {\displaystyle x=X\cos(\omega t+\varphi )} , where X {\displaystyle X} is the amplitude, and φ {\displaystyle \varphi } is the phase angle. Applying the virial theorem, we have m ⟨ x ˙ x ˙ ⟩ = k ⟨ x x ⟩ + γ ⟨ x x ˙ ⟩ − F ⟨ cos ⁡ ( ω t ) x ⟩ {\displaystyle m\langle {\dot {x}}{\dot {x}}\rangle =k\langle xx\rangle +\gamma \langle x{\dot {x}}\rangle -F\langle \cos(\omega t)x\rangle } , which simplifies to F cos ⁡ ( φ ) = m ( ω 0 2 − ω 2 ) X {\displaystyle F\cos(\varphi )=m(\omega _{0}^{2}-\omega ^{2})X} , where ω 0 = k / m {\displaystyle \omega _{0}={\sqrt {k/m}}}

6790-1439: The particles are at diametrically opposite points of a circular orbit with radius r . The velocities are v 1 ( t ) and v 2 ( t ) = − v 1 ( t ) , which are normal to forces F 1 ( t ) and F 2 ( t ) = − F 1 ( t ) . The respective magnitudes are fixed at v and F . The average kinetic energy of the system in an interval of time from t 1 to t 2 is ⟨ T ⟩ = 1 t 2 − t 1 ∫ t 1 t 2 ∑ k = 1 N 1 2 m k | v k ( t ) | 2 d t = 1 t 2 − t 1 ∫ t 1 t 2 ( 1 2 m | v 1 ( t ) | 2 + 1 2 m | v 2 ( t ) | 2 ) d t = m v 2 . {\displaystyle \langle T\rangle ={\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}{\frac {1}{2}}m_{k}|\mathbf {v} _{k}(t)|^{2}\,dt={\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}\left({\frac {1}{2}}m|\mathbf {v} _{1}(t)|^{2}+{\frac {1}{2}}m|\mathbf {v} _{2}(t)|^{2}\right)\,dt=mv^{2}.} Taking center of mass as

6887-920: The position operator X n and the momentum operator P n = − i ℏ d d X n {\displaystyle P_{n}=-i\hbar {\frac {d}{dX_{n}}}} of particle n , [ H , X n P n ] = X n [ H , P n ] + [ H , X n ] P n = i ℏ X n d V d X n − i ℏ P n 2 m n . {\displaystyle [H,X_{n}P_{n}]=X_{n}[H,P_{n}]+[H,X_{n}]P_{n}=i\hbar X_{n}{\frac {dV}{dX_{n}}}-i\hbar {\frac {P_{n}^{2}}{m_{n}}}.} Summing over all particles, one finds that for Q = ∑ n X n P n {\displaystyle Q=\sum _{n}X_{n}P_{n}}

6984-427: 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

7081-570: 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,

7178-471: 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

7275-477: 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

7372-435: The ratio of kinetic energy to potential energy is no longer fixed, but necessarily falls into an interval: 2 ⟨ T TOT ⟩ n ⟨ V TOT ⟩ ∈ [ 1 , 2 ] , {\displaystyle {\frac {2\langle T_{\text{TOT}}\rangle }{n\langle V_{\text{TOT}}\rangle }}\in [1,2],} where the more relativistic systems exhibit

7469-432: 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

7566-408: 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,

7663-413: 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

7760-445: 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

7857-706: The solution { X = F 2 γ 2 ω 2 + m 2 ( ω 0 2 − ω 2 ) 2 , tan ⁡ φ = − γ ω m ( ω 0 2 − ω 2 ) . {\displaystyle {\begin{cases}X={\sqrt {\dfrac {F^{2}}{\gamma ^{2}\omega ^{2}+m^{2}(\omega _{0}^{2}-\omega ^{2})^{2}}}},\\\tan \varphi =-{\dfrac {\gamma \omega }{m(\omega _{0}^{2}-\omega ^{2})}}.\end{cases}}} Consider

7954-419: 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

8051-460: 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

8148-1777: The sum in terms below and above this diagonal and add them together in pairs: ∑ k = 1 N F k ⋅ r k = ∑ k = 1 N ∑ j = 1 N F j k ⋅ r k = ∑ k = 2 N ∑ j = 1 k − 1 ( F j k ⋅ r k + F k j ⋅ r j ) = ∑ k = 2 N ∑ j = 1 k − 1 ( F j k ⋅ r k − F j k ⋅ r j ) = ∑ k = 2 N ∑ j = 1 k − 1 F j k ⋅ ( r k − r j ) , {\displaystyle {\begin{aligned}\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}&=\sum _{k=1}^{N}\sum _{j=1}^{N}\mathbf {F} _{jk}\cdot \mathbf {r} _{k}=\sum _{k=2}^{N}\sum _{j=1}^{k-1}(\mathbf {F} _{jk}\cdot \mathbf {r} _{k}+\mathbf {F} _{kj}\cdot \mathbf {r} _{j})\\&=\sum _{k=2}^{N}\sum _{j=1}^{k-1}(\mathbf {F} _{jk}\cdot \mathbf {r} _{k}-\mathbf {F} _{jk}\cdot \mathbf {r} _{j})=\sum _{k=2}^{N}\sum _{j=1}^{k-1}\mathbf {F} _{jk}\cdot (\mathbf {r} _{k}-\mathbf {r} _{j}),\end{aligned}}} where we have used Newton's third law of motion , i.e., F jk = − F kj (equal and opposite reaction). It often happens that

8245-739: The system under consideration, the averaging need not be taken over time; an ensemble average can also be taken, with equivalent results. Although originally derived for classical mechanics, the virial theorem also holds for quantum mechanics, as first shown by Fock using the Ehrenfest theorem . Evaluate the commutator of the Hamiltonian H = V ( { X i } ) + ∑ n P n 2 2 m n {\displaystyle H=V{\bigl (}\{X_{i}\}{\bigr )}+\sum _{n}{\frac {P_{n}^{2}}{2m_{n}}}} with

8342-462: 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

8439-413: 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

8536-403: 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

8633-1273: The time derivative of G is d G d t = ∑ k = 1 N p k ⋅ d r k d t + ∑ k = 1 N d p k d t ⋅ r k = ∑ k = 1 N m k d r k d t ⋅ d r k d t + ∑ k = 1 N F k ⋅ r k = 2 T + ∑ k = 1 N F k ⋅ r k , {\displaystyle {\begin{aligned}{\frac {dG}{dt}}&=\sum _{k=1}^{N}\mathbf {p} _{k}\cdot {\frac {d\mathbf {r} _{k}}{dt}}+\sum _{k=1}^{N}{\frac {d\mathbf {p} _{k}}{dt}}\cdot \mathbf {r} _{k}\\&=\sum _{k=1}^{N}m_{k}{\frac {d\mathbf {r} _{k}}{dt}}\cdot {\frac {d\mathbf {r} _{k}}{dt}}+\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}\\&=2T+\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k},\end{aligned}}} where m k

8730-494: The total mass is 450 times that of observed mass. Lord Rayleigh published a generalization of the virial theorem in 1900, which was partially reprinted in 1903. Henri Poincaré proved and applied a form of the virial theorem in 1911 to the problem of formation of the Solar System from a proto-stellar cloud (then known as cosmogony ). A variational form of the virial theorem was developed in 1945 by Ledoux. A tensor form of

8827-428: The virial theorem concerns galaxy clusters . If a region of space is unusually full of galaxies, it is safe to assume that they have been together for a long time, and the virial theorem can be applied. Doppler effect measurements give lower bounds for their relative velocities, and the virial theorem gives a lower bound for the total mass of the cluster, including any dark matter. If the ergodic hypothesis holds for

8924-710: The virial theorem to estimate the mass of Coma Cluster , and discovered a discrepancy of mass of about 450, which he explained as due to "dark matter". He refined the analysis in 1937, finding a discrepancy of about 500. He approximated the Coma cluster as a spherical "gas" of N {\displaystyle N} stars of roughly equal mass m {\displaystyle m} , which gives ⟨ T ⟩ = 1 2 N m ⟨ v 2 ⟩ {\textstyle \langle T\rangle ={\frac {1}{2}}Nm\langle v^{2}\rangle } . The total gravitational potential energy of

9021-954: The virial theorem was developed by Parker, Chandrasekhar and Fermi. The following generalization of the virial theorem has been established by Pollard in 1964 for the case of the inverse square law: 2 lim τ → + ∞ ⟨ T ⟩ τ = lim τ → + ∞ ⟨ U ⟩ τ if and only if lim τ → + ∞ τ − 2 I ( τ ) = 0. {\displaystyle 2\lim _{\tau \to +\infty }\langle T\rangle _{\tau }=\lim _{\tau \to +\infty }\langle U\rangle _{\tau }\quad {\text{if and only if}}\quad \lim _{\tau \to +\infty }{\tau }^{-2}I(\tau )=0.} A boundary term otherwise must be added. The virial theorem can be extended to include electric and magnetic fields. The result

9118-520: The virial theorem, the total mass of the cluster should be 5 R ⟨ v r 2 ⟩ G ≈ 3.6 × 10 14 M ⊙ {\displaystyle {\frac {5R\langle v_{r}^{2}\rangle }{G}}\approx 3.6\times 10^{14}M_{\odot }} . However, the observed mass is N m = 8 × 10 11 M ⊙ {\displaystyle Nm=8\times 10^{11}M_{\odot }} , meaning

9215-446: 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

9312-496: 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"

9409-597: Was the first to use the virial theorem to deduce the existence of unseen matter, which is now called dark matter . Richard Bader showed that the charge distribution of a total system can be partitioned into its kinetic and potential energies that obey the virial theorem. As another example of its many applications, the virial theorem has been used to derive the Chandrasekhar limit for the stability of white dwarf stars . Consider N = 2 particles with equal mass m , acted upon by mutually attractive forces. Suppose

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