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132-408: The equipartition theorem makes quantitative predictions. Like the virial theorem , it gives the total average kinetic and potential energies for a system at a given temperature, from which the system's heat capacity can be computed. However, equipartition also gives the average values of individual components of the energy, such as the kinetic energy of a particular particle or the potential energy of

264-433: A spring , which has a quadratic potential energy where the constant a describes the stiffness of the spring and q is the deviation from equilibrium. If such a one-dimensional system has mass m , then its kinetic energy H kin is where v and p = mv denote the velocity and momentum of the oscillator. Combining these terms yields the total energy Equipartition therefore implies that in thermal equilibrium,

396-468: A statistical point of view ). Therefore, if such systems have equal temperatures, they are at thermal equilibrium . However, this equilibrium is stable only if the systems have positive heat capacities. For such systems, when heat flows from a higher-temperature system to a lower-temperature one, the temperature of the first decreases and that of the latter increases, so that both approach equilibrium. In contrast, for systems with negative heat capacities,

528-442: 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

660-1752: 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}}

792-405: A component of the position or velocity of a particle) which appears only quadratically in the energy has an average energy of 1 ⁄ 2 k B T and therefore contributes 1 ⁄ 2 k B to the system's heat capacity . This has many applications. The (Newtonian) kinetic energy of a particle of mass m , velocity v is given by where v x , v y and v z are

924-596: A constant volume ( d V = 0 {\displaystyle dV=0} ) the heat capacity reads: C V = δ Q d T | V = const = ( ∂ U ∂ T ) V {\displaystyle C_{V}={\frac {\delta Q}{dT}}{\Bigr |}_{V={\text{const}}}=\left({\frac {\partial U}{\partial T}}\right)_{V}} The relation between C V {\displaystyle C_{V}} and C p {\displaystyle C_{p}}

1056-445: 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} }}}

1188-466: A degree of freedom x n appears only as a quadratic term a n x n in the Hamiltonian H , then the first of these formulae implies that which is twice the contribution that this degree of freedom makes to the average energy ⟨ H ⟩ {\displaystyle \langle H\rangle } . Thus the equipartition theorem for systems with quadratic energies follows easily from

1320-692: 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

1452-456: A function C ( p , T ) {\displaystyle C(p,T)} of those two variables. The variation can be ignored in contexts when working with objects in narrow ranges of temperature and pressure. For example, the heat capacity of a block of iron weighing one pound is about 204 J/K when measured from a starting temperature T  = 25 °C and P  = 1 atm of pressure. That approximate value

SECTION 10

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1584-412: 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 } ,

1716-475: A single spring . For example, it predicts that every atom in a monatomic ideal gas has an average kinetic energy of ⁠ 3 / 2 ⁠ k B T in thermal equilibrium, where k B is the Boltzmann constant and T is the (thermodynamic) temperature . More generally, equipartition can be applied to any classical system in thermal equilibrium , no matter how complicated. It can be used to derive

1848-1589: 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

1980-730: 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

2112-509: 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

2244-413: A time average of a single system. The general equipartition theorem holds in both the microcanonical ensemble , when the total energy of the system is constant, and also in the canonical ensemble , when the system is coupled to a heat bath with which it can exchange energy. Derivations of the general formula are given later in the article . The general formula is equivalent to the following two: If

2376-672: 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

2508-518: Is negative . Examples include a reversibly and nearly adiabatically expanding ideal gas, which cools, Δ T {\displaystyle \Delta T} < 0, while a small amount of heat Q {\displaystyle Q} > 0 is put in, or combusting methane with increasing temperature, Δ T {\displaystyle \Delta T} > 0, and giving off heat, Q {\displaystyle Q} < 0. Others are inhomogeneous systems that do not meet

2640-418: Is a physical property of matter , defined as the amount of heat to be supplied to an object to produce a unit change in its temperature . The SI unit of heat capacity is joule per kelvin (J/K). Heat capacity is an extensive property . The corresponding intensive property is the specific heat capacity , found by dividing the heat capacity of an object by its mass. Dividing the heat capacity by

2772-428: Is adequate for temperatures between 15 °C and 35 °C, and surrounding pressures from 0 to 10 atmospheres, because the exact value varies very little in those ranges. One can trust that the same heat input of 204 J will raise the temperature of the block from 15 °C to 16 °C, or from 34 °C to 35 °C, with negligible error. At constant pressure, heat supplied to the system contributes to both

SECTION 20

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2904-794: Is always less than the value of C p . {\displaystyle C_{p}.} ( C V {\displaystyle C_{V}} < C p . {\displaystyle C_{p}.} ) Expressing the inner energy as a function of the variables T {\displaystyle T} and V {\displaystyle V} gives: δ Q = ( ∂ U ∂ T ) V d T + ( ∂ U ∂ V ) T d V + p d V {\displaystyle \delta Q=\left({\frac {\partial U}{\partial T}}\right)_{V}dT+\left({\frac {\partial U}{\partial V}}\right)_{T}dV+pdV} For

3036-2126: 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

3168-512: Is equal to 1 ⁄ 2 (mass)(velocity), the heavier atoms of xenon have a lower average speed than do the lighter atoms of helium at the same temperature. Figure 2 shows the Maxwell–Boltzmann distribution for the speeds of the atoms in four noble gases. In this example, the key point is that the kinetic energy is quadratic in the velocity. The equipartition theorem shows that in thermal equilibrium, any degree of freedom (such as

3300-543: Is inaccurate at lower temperatures, due to quantum effects; it is also inconsistent with the experimentally derived third law of thermodynamics , according to which the molar heat capacity of any substance must go to zero as the temperature goes to absolute zero. A more accurate theory, incorporating quantum effects, was developed by Albert Einstein (1907) and Peter Debye (1911). Many other physical systems can be modeled as sets of coupled oscillators . The motions of such oscillators can be decomposed into normal modes , like

3432-587: Is joule per kelvin (J/K or J⋅K ). Since an increment of temperature of one degree Celsius is the same as an increment of one kelvin, that is the same unit as J/°C. The heat capacity of an object is an amount of energy divided by a temperature change, which has the dimension L ⋅M⋅T ⋅Θ . Therefore, the SI unit J/K is equivalent to kilogram meter squared per second squared per kelvin (kg⋅m ⋅s ⋅K ). Professionals in construction , civil engineering , chemical engineering , and other technical disciplines, especially in

3564-507: 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

3696-432: Is the gas constant . Since R ≈ 2 cal /( mol · K ), equipartition predicts that the molar heat capacity of an ideal gas is roughly 3 cal/(mol·K). This prediction is confirmed by experiment when compared to monatomic gases. The mean kinetic energy also allows the root mean square speed v rms of the gas particles to be calculated: where M = N A m is the mass of a mole of gas particles. This result

3828-879: 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 },} Heat capacity Heat capacity or thermal capacity

3960-605: Is the amount of heat that must be added to the object (of mass M ) in order to raise its temperature by Δ T {\displaystyle \Delta T} . The value of this parameter usually varies considerably depending on the starting temperature T {\displaystyle T} of the object and the pressure p {\displaystyle p} applied to it. In particular, it typically varies dramatically with phase transitions such as melting or vaporization (see enthalpy of fusion and enthalpy of vaporization ). Therefore, it should be considered

4092-627: 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

Equipartition theorem - Misplaced Pages Continue

4224-844: 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

4356-940: 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

4488-607: Is the number of degrees of freedom of each individual particle in the gas, and N i = N f − 3 {\displaystyle N_{i}=N_{f}-3} is the number of internal degrees of freedom , where the number 3 comes from the three translational degrees of freedom (for a gas in 3D space). This means that a monoatomic ideal gas (with zero internal degrees of freedom) will have isochoric heat capacity C v = 3 n R 2 {\displaystyle C_{v}={\frac {3nR}{2}}} . No change in internal energy (as

4620-993: 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

4752-492: Is the total kinetic energy of the N particles, F k represents the force on the k th particle, which is located at position r k , and angle brackets represent the average over time of the enclosed quantity. The word virial for the right-hand side of the equation derives from vis , the Latin word for "force" or "energy", and was given its technical definition by Rudolf Clausius in 1870. The significance of

4884-425: 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

5016-595: 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

5148-446: Is then: C p = C V + ( ( ∂ U ∂ V ) T + p ) ( ∂ V ∂ T ) p {\displaystyle C_{p}=C_{V}+\left(\left({\frac {\partial U}{\partial V}}\right)_{T}+p\right)\left({\frac {\partial V}{\partial T}}\right)_{p}} Mayer's relation : where: Using

5280-416: Is useful for many applications such as Graham's law of effusion , which provides a method for enriching uranium . A similar example is provided by a rotating molecule with principal moments of inertia I 1 , I 2 and I 3 . According to classical mechanics, the rotational energy of such a molecule is given by where ω 1 , ω 2 , and ω 3 are the principal components of

5412-524: The United States , may use the so-called English Engineering units , that include the pound (lb = 0.45359237 kg) as the unit of mass, the degree Fahrenheit or Rankine ( ⁠ 5 / 9 ⁠ K, about 0.55556 K) as the unit of temperature increment, and the British thermal unit (BTU ≈ 1055.06 J), as the unit of heat. In those contexts, the unit of heat capacity is 1 BTU/°R ≈ 1900 J/K. The BTU

Equipartition theorem - Misplaced Pages Continue

5544-412: The angular velocity . By exactly the same reasoning as in the translational case, equipartition implies that in thermal equilibrium the average rotational energy of each particle is ⁠ 3 / 2 ⁠ k B T . Similarly, the equipartition theorem allows the average (more precisely, the root mean square) angular speed of the molecules to be calculated. The tumbling of rigid molecules—that is,

5676-817: 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

5808-1068: The first law of thermodynamics follows δ Q = d U + p d V {\displaystyle \delta Q=dU+pdV} and the inner energy as a function of p {\displaystyle p} and T {\displaystyle T} is: δ Q = ( ∂ U ∂ T ) p d T + ( ∂ U ∂ p ) T d p + p [ ( ∂ V ∂ T ) p d T + ( ∂ V ∂ p ) T d p ] {\displaystyle \delta Q=\left({\frac {\partial U}{\partial T}}\right)_{p}dT+\left({\frac {\partial U}{\partial p}}\right)_{T}dp+p\left[\left({\frac {\partial V}{\partial T}}\right)_{p}dT+\left({\frac {\partial V}{\partial p}}\right)_{T}dp\right]} For constant pressure ( d p = 0 ) {\displaystyle (dp=0)}

5940-511: The gas constant R and the Boltzmann constant k B , this provides an explanation for the Dulong–Petit law of specific heat capacities of solids, which stated that the specific heat capacity (per unit mass) of a solid element is inversely proportional to its atomic weight . A modern version is that the molar heat capacity of a solid is 3R  ≈ 6 cal/(mol·K). However, this law

6072-448: The ideal gas law from classical mechanics. If q = ( q x , q y , q z ) and p = ( p x , p y , p z ) denote the position vector and momentum of a particle in the gas, and F is the net force on that particle, then where the first equality is Newton's second law , and the second line uses Hamilton's equations and the equipartition formula. Summing over a system of N particles yields By Newton's third law and

6204-458: The ideal gas law , and the Dulong–Petit law for the specific heat capacities of solids. The equipartition theorem can also be used to predict the properties of stars , even white dwarfs and neutron stars , since it holds even when relativistic effects are considered. Although the equipartition theorem makes accurate predictions in certain conditions, it is inaccurate when quantum effects are significant, such as at low temperatures. When

6336-1026: 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,

6468-444: 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

6600-402: The temperature T . Another difference is that traditional derivations of the virial theorem use averages over time, whereas those of the equipartition theorem use averages over phase space . Ideal gases provide an important application of the equipartition theorem. As well as providing the formula for the average kinetic energy per particle, the equipartition theorem can be used to derive

6732-564: The thermal energy k B T is smaller than the quantum energy spacing in a particular degree of freedom , the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition. Such a degree of freedom is said to be "frozen out" when the thermal energy is much smaller than this spacing. For example, the heat capacity of a solid decreases at low temperatures as various types of motion become frozen out, rather than remaining constant as predicted by equipartition. Such decreases in heat capacity were among

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6864-426: The work done and the change in internal energy , according to the first law of thermodynamics . The heat capacity is called C p {\displaystyle C_{p}} and defined as: C p = δ Q d T | p = c o n s t {\displaystyle C_{p}={\frac {\delta Q}{dT}}{\Bigr |}_{p=const}} From

6996-510: The Cartesian components of the velocity v . Here, H is short for Hamiltonian , and used henceforth as a symbol for energy because the Hamiltonian formalism plays a central role in the most general form of the equipartition theorem. Since the kinetic energy is quadratic in the components of the velocity, by equipartition these three components each contribute 1 ⁄ 2 k B T to

7128-568: The French physicists Pierre Louis Dulong and Alexis Thérèse Petit discovered that the specific heat capacities of solid elements at room temperature were inversely proportional to the atomic weight of the element. Their law was used for many years as a technique for measuring atomic weights. However, subsequent studies by James Dewar and Heinrich Friedrich Weber showed that this Dulong–Petit law holds only at high temperatures ; at lower temperatures, or for exceptionally hard solids such as diamond ,

7260-460: The above two relations, the specific heats can be deduced as follows: Following from the equipartition of energy , it is deduced that an ideal gas has the isochoric heat capacity C V = n R N f 2 = n R 3 + N i 2 {\displaystyle C_{V}=nR{\frac {N_{f}}{2}}=nR{\frac {3+N_{i}}{2}}} where N f {\displaystyle N_{f}}

7392-433: The amount of substance in moles yields its molar heat capacity . The volumetric heat capacity measures the heat capacity per volume . In architecture and civil engineering , the heat capacity of a building is often referred to as its thermal mass . The heat capacity of an object, denoted by C {\displaystyle C} , is the limit where Δ Q {\displaystyle \Delta Q}

7524-415: The average , once the system has reached thermal equilibrium. Equipartition also makes quantitative predictions for these energies. For example, it predicts that every atom of an inert noble gas , in thermal equilibrium at temperature T , has an average translational kinetic energy of ⁠ 3 / 2 ⁠ k B T , where k B is the Boltzmann constant . As a consequence, since kinetic energy

7656-405: The average energy was divided equally among all the independent components of motion in a system. Boltzmann applied the equipartition theorem to provide a theoretical explanation of the Dulong–Petit law for the specific heat capacities of solids. The history of the equipartition theorem is intertwined with that of specific heat capacity , both of which were studied in the 19th century. In 1819,

7788-496: The average kinetic energy U kin are locked together in the relation The total energy U (= U pot + U kin ) therefore obeys If the system loses energy, for example, by radiating energy into space, the average kinetic energy actually increases. If a temperature is defined by the average kinetic energy, then the system therefore can be said to have a negative heat capacity. A more extreme version of this occurs with black holes . According to black-hole thermodynamics ,

7920-492: 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

8052-436: The average kinetic energy in thermal equilibrium. Thus the average kinetic energy of the particle is ⁠ 3 / 2 ⁠ k B T , as in the example of noble gases above. More generally, in a monatomic ideal gas the total energy consists purely of (translational) kinetic energy: by assumption, the particles have no internal degrees of freedom and move independently of one another. Equipartition therefore predicts that

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8184-1196: 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

8316-797: 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

8448-626: The average over time of the total kinetic energy of a stable system of discrete particles, bound by a conservative force (where the work done is independent of path) with that of the total potential energy of the system. Mathematically, the theorem states that ⟨ T ⟩ = − 1 2 ∑ k = 1 N ⟨ F k ⋅ r k ⟩ , {\displaystyle \langle T\rangle =-{\frac {1}{2}}\,\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle ,} where T

8580-425: The average position of a particular clump of buoyant mass m b . For an infinitely tall bottle of beer, the gravitational potential energy is given by where z is the height of the protein clump in the bottle and g is the acceleration due to gravity. Since s = 1 , the average potential energy of a protein clump equals k B T . Hence, a protein clump with a buoyant mass of 10  MDa (roughly

8712-411: 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

8844-524: The classical Drude model , metallic electrons act as a nearly ideal gas, and so they should contribute ⁠ 3 / 2 ⁠ N e k B to the heat capacity by the equipartition theorem, where N e is the number of electrons. Experimentally, however, electrons contribute little to the heat capacity: the molar heat capacities of many conductors and insulators are nearly the same. Several explanations of equipartition's failure to account for molar heat capacities were proposed. Boltzmann defended

8976-514: 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

9108-531: 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

9240-843: 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

9372-453: 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}}

9504-1075: 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,

9636-422: The derivation of his equipartition theorem as correct, but suggested that gases might not be in thermal equilibrium because of their interactions with the aether . Lord Kelvin suggested that the derivation of the equipartition theorem must be incorrect, since it disagreed with experiment, but was unable to show how. In 1900 Lord Rayleigh instead put forward a more radical view that the equipartition theorem and

9768-405: The development of quantum mechanics and quantum field theory . The name "equipartition" means "equal division," as derived from the Latin equi from the antecedent, æquus ("equal or even"), and partition from the noun, partitio ("division, portion"). The original concept of equipartition was that the total kinetic energy of a system is shared equally among all of its independent parts, on

9900-425: The disagreement between experiment and the equipartition theorem was much worse than even these numbers suggest; since atoms have internal parts, heat energy should go into the motion of these internal parts, making the predicted specific heats of monatomic and diatomic gases much higher than 3 cal/(mol·K) and 7 cal/(mol·K), respectively. A third discrepancy concerned the specific heat of metals. According to

10032-468: The effective heat capacity of the gas, in that situation, will have a value intermediate between its isobaric and isochoric capacities C p {\displaystyle C_{p}} and C V {\displaystyle C_{V}} . For complex thermodynamic systems with several interacting parts and state variables , or for measurement conditions that are neither constant pressure nor constant volume, or for situations where

10164-844: 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

10296-655: The equation simplifies to: C p = δ Q d T | p = c o n s t = ( ∂ U ∂ T ) p + p ( ∂ V ∂ T ) p = ( ∂ H ∂ T ) p {\displaystyle C_{p}={\frac {\delta Q}{dT}}{\Bigr |}_{p=const}=\left({\frac {\partial U}{\partial T}}\right)_{p}+p\left({\frac {\partial V}{\partial T}}\right)_{p}=\left({\frac {\partial H}{\partial T}}\right)_{p}} where

10428-417: The equations of Hamiltonian mechanics , these formulae may also be written Similarly, one can show using formula 2 that and The general equipartition theorem is an extension of the virial theorem (proposed in 1870), which states that where t denotes time . Two key differences are that the virial theorem relates summed rather than individual averages to each other, and it does not connect them to

10560-1007: 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

10692-415: The experimental assumption of thermal equilibrium were both correct; to reconcile them, he noted the need for a new principle that would provide an "escape from the destructive simplicity" of the equipartition theorem. Albert Einstein provided that escape, by showing in 1906 that these anomalies in the specific heat were due to quantum effects, specifically the quantization of energy in the elastic modes of

10824-449: 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

10956-1307: 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

11088-485: The final equality follows from the appropriate Maxwell relations , and is commonly used as the definition of the isobaric heat capacity. A system undergoing a process at constant volume implies that no expansion work is done, so the heat supplied contributes only to the change in internal energy. The heat capacity obtained this way is denoted C V . {\displaystyle C_{V}.} The value of C V {\displaystyle C_{V}}

11220-420: The first signs to physicists of the 19th century that classical physics was incorrect and that a new, more subtle, scientific model was required. Along with other evidence, equipartition's failure to model black-body radiation —also known as the ultraviolet catastrophe —led Max Planck to suggest that energy in the oscillators in an object, which emit light, were quantized, a revolutionary hypothesis that spurred

11352-484: The following equipartition formula holds in thermal equilibrium for all indices m and n : Here δ mn is the Kronecker delta , which is equal to one if m = n and is zero otherwise. The averaging brackets ⟨ … ⟩ {\displaystyle \left\langle \ldots \right\rangle } is assumed to be an ensemble average over phase space or, under an assumption of ergodicity ,

11484-746: 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

11616-450: The general formula. A similar argument, with 2 replaced by s , applies to energies of the form a n x n . The degrees of freedom x n are coordinates on the phase space of the system and are therefore commonly subdivided into generalized position coordinates q k and generalized momentum coordinates p k , where p k is the conjugate momentum to q k . In this situation, formula 1 means that for all k , Using

11748-771: 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

11880-442: The haze sometimes seen in beer can be caused by clumps of proteins that scatter light. Over time, these clumps settle downwards under the influence of gravity, causing more haze near the bottom of a bottle than near its top. However, in a process working in the opposite direction, the particles also diffuse back up towards the top of the bottle. Once equilibrium has been reached, the equipartition theorem may be used to determine

12012-426: The ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure P of the gas. Hence where d S is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is Virial theorem In statistical mechanics , the virial theorem provides a general equation that relates

12144-399: The individual parts. However, this computation is valid only when all parts of the object are at the same external pressure before and after the measurement. That may not be possible in some cases. For example, when heating an amount of gas in an elastic container, its volume and pressure will both increase, even if the atmospheric pressure outside the container is kept constant. Therefore,

12276-462: The initial and final states. Namely, one must somehow specify how the positions, velocities, pressures, volumes, etc. changed between the initial and final states; and use the general tools of thermodynamics to predict the system's reaction to a small energy input. The "constant volume" and "constant pressure" heating modes are just two among infinitely many paths that a simple homogeneous system can follow. The heat capacity can usually be measured by

12408-881: 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

12540-458: The law of equipartition to hold. Potential energies are not always quadratic in the position. However, the equipartition theorem also shows that if a degree of freedom x contributes only a multiple of x (for a fixed real number s ) to the energy, then in thermal equilibrium the average energy of that part is k B T / s . There is a simple application of this extension to the sedimentation of particles under gravity . For example,

12672-399: 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

12804-405: The method implied by its definition: start with the object at a known uniform temperature, add a known amount of heat energy to it, wait for its temperature to become uniform, and measure the change in its temperature. This method can give moderately accurate values for many solids; however, it cannot provide very precise measurements, especially for gases. The SI unit for heat capacity of an object

12936-427: The more mass and energy a black hole absorbs, the colder it becomes. In contrast, if it is a net emitter of energy, through Hawking radiation , it will become hotter and hotter until it boils away. According to the second law of thermodynamics , when two systems with different temperatures interact via a purely thermal connection, heat will flow from the hotter system to the cooler one (this can also be understood from

13068-456: 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

13200-403: 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,

13332-951: 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

13464-539: The oscillator has average energy where the angular brackets ⟨ … ⟩ {\displaystyle \left\langle \ldots \right\rangle } denote the average of the enclosed quantity, This result is valid for any type of harmonic oscillator, such as a pendulum , a vibrating molecule or a passive electronic oscillator . Systems of such oscillators arise in many situations; by equipartition, each such oscillator receives an average total energy k B T and hence contributes k B to

13596-1272: 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}}}

13728-1439: 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

13860-920: 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}}

13992-524: The random rotations of molecules in solution—plays a key role in the relaxations observed by nuclear magnetic resonance , particularly protein NMR and residual dipolar couplings . Rotational diffusion can also be observed by other biophysical probes such as fluorescence anisotropy , flow birefringence and dielectric spectroscopy . Equipartition applies to potential energies as well as kinetic energies: important examples include harmonic oscillators such as

14124-435: 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

14256-578: The size of a virus ) would produce a haze with an average height of about 2 cm at equilibrium. The process of such sedimentation to equilibrium is described by the Mason–Weaver equation . The equipartition of kinetic energy was proposed initially in 1843, and more correctly in 1845, by John James Waterston . In 1859, James Clerk Maxwell argued that the kinetic heat energy of a gas is equally divided between linear and rotational energy. In 1876, Ludwig Boltzmann expanded on this principle by showing that

14388-488: The solid can be viewed as a system of 3 N independent simple harmonic oscillators , where N denotes the number of atoms in the lattice. Since each harmonic oscillator has average energy k B T , the average total energy of the solid is 3 N   k B T , and its heat capacity is 3 N   k B . By taking N to be the Avogadro constant N A , and using the relation R = N A k B between

14520-490: The solid. Einstein used the failure of equipartition to argue for the need of a new quantum theory of matter. Nernst's 1910 measurements of specific heats at low temperatures supported Einstein's theory, and led to the widespread acceptance of quantum theory among physicists. The most general form of the equipartition theorem states that under suitable assumptions (discussed below ), for a physical system with Hamiltonian energy function H and degrees of freedom x n ,

14652-706: 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

14784-592: The specific heat capacity was lower. Experimental observations of the specific heat capacities of gases also raised concerns about the validity of the equipartition theorem. The theorem predicts that the molar heat capacity of simple monatomic gases should be roughly 3 cal/(mol·K), whereas that of diatomic gases should be roughly 7 cal/(mol·K). Experiments confirmed the former prediction, but found that molar heat capacities of diatomic gases were typically about 5 cal/(mol·K), and fell to about 3 cal/(mol·K) at very low temperatures. Maxwell noted in 1875 that

14916-408: The state of the material rather than raising the overall temperature. The heat capacity may be well-defined even for heterogeneous objects, with separate parts made of different materials; such as an electric motor , a crucible with some metal, or a whole building. In many cases, the (isobaric) heat capacity of such objects can be computed by simply adding together the (isobaric) heat capacities of

15048-417: The strict definition of thermodynamic equilibrium. They include gravitating objects such as stars and galaxies, and also some nano-scale clusters of a few tens of atoms close to a phase transition. A negative heat capacity can result in a negative temperature . According to the virial theorem , for a self-gravitating body like a star or an interstellar gas cloud, the average potential energy U pot and

15180-1777: 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

15312-676: 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

15444-423: The system's heat capacity . This can be used to derive the formula for Johnson–Nyquist noise and the Dulong–Petit law of solid heat capacities. The latter application was particularly significant in the history of equipartition. An important application of the equipartition theorem is to the specific heat capacity of a crystalline solid. Each atom in such a solid can oscillate in three independent directions, so

15576-430: The temperature is significantly non-uniform, the simple definitions of heat capacity above are not useful or even meaningful. The heat energy that is supplied may end up as kinetic energy (energy of motion) and potential energy (energy stored in force fields), both at macroscopic and atomic scales. Then the change in temperature will depend on the particular path that the system followed through its phase space between

15708-454: The temperature of the hotter system will further increase as it loses heat, and that of the colder will further decrease, so that they will move farther from equilibrium. This means that the equilibrium is unstable . For example, according to theory, the smaller (less massive) a black hole is, the smaller its Schwarzschild radius will be, and therefore the greater the curvature of its event horizon will be, as well as its temperature. Thus,

15840-406: The temperature of the system is constant throughout the process) leads to only work done by the total supplied heat, and thus an infinite amount of heat is required to increase the temperature of the system by a unit temperature, leading to infinite or undefined heat capacity of the system. Heat capacity of a system undergoing phase transition is infinite , because the heat is utilized in changing

15972-1273: 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

16104-424: The total energy of an ideal gas of N particles is ⁠ 3 / 2 ⁠ N   k B   T . It follows that the heat capacity of the gas is ⁠ 3 / 2 ⁠ N   k B and hence, in particular, the heat capacity of a mole of such gas particles is ⁠ 3 / 2 ⁠ N A k B = ⁠ 3 / 2 ⁠ R , where N A is the Avogadro constant and R

16236-492: 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

16368-443: The units of heat capacity are Most physical systems exhibit a positive heat capacity; constant-volume and constant-pressure heat capacities, rigorously defined as partial derivatives, are always positive for homogeneous bodies. However, even though it can seem paradoxical at first, there are some systems for which the heat capacity Q {\displaystyle Q} / Δ T {\displaystyle \Delta T}

16500-415: The vibrational modes of a piano string or the resonances of an organ pipe . On the other hand, equipartition often breaks down for such systems, because there is no exchange of energy between the normal modes. In an extreme situation, the modes are independent and so their energies are independently conserved. This shows that some sort of mixing of energies, formally called ergodicity , is important for

16632-428: 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

16764-526: 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

16896-709: 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

17028-961: 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

17160-520: 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

17292-428: Was in fact defined so that the average heat capacity of one pound of water would be 1 BTU/°F. In this regard, with respect to mass, note conversion of 1 Btu/lb⋅°R ≈ 4,187 J/kg⋅K and the calorie (below). In chemistry, heat amounts are often measured in calories . Confusingly, two units with that name, denoted "cal" or "Cal", have been commonly used to measure amounts of heat: With these units of heat energy,

17424-591: 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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