Two Lectures About the Virial Theorem

Paolo Podio-Guidugli Accademia Nazionale dei Lincei and Department of Mathematics, Universita` di Roma TorVergata

Two Lectures about the Virial Theorem

HMM Summer School at OIST - Okinawa, Japan 7 and 8 April 2017 PLAN Lecture I - Deterministic Virial

I.1 Discrete and Newtonian I.2 Continuum and Eulerian

Lecture II - Probabilistic Virial

II.1 Recap of Statistical Mechanics II.2 The Equipartition Theorem II.3 The Virial Theorem Lecture I - Deterministic Virial

I.1 Discrete and Newtonian – Ordinary forces – Fluctuating forces – Particle systems I.2 Continuum and Eulerian

Main reference: G. Marc and W.G. McMillan, The virial theorem. Adv. Chem. Phys. 58, 209-361 (1985). I.1 Deterministic Virial, Discrete and Newtonian Ordinary forces

“The mean vis viva [ kinetic energy] of the system is equal to its ⌘ [force] virial” (Rudolf Clausius, 1870)

A[f x]= 2 A[k], 2 k := mx˙ x˙ mx¨ = f , x = x(t) o · · ( mx¨ = f mx¨ x = f x =: the virial of force f . ) · · Note that d (x˙ x) x˙ x˙ dt · · x¨ x = 8 . · > 2 > 1 d < (x x) x˙ x˙ 2 dt2 · · > With a use of upper alternative,:> d (mx˙ x) mx˙ x˙ = f x, dt · · · or rather, on setting w := mx˙ x and 2 k := mx˙ x˙ , · · f x =˙w 2k. · Introduce the time-average operator 1 ⌧ A[f]:= lim f(t)dt . ⌧ + ⌧ 7! 1 Z0 Then, for f(t)=g˙(t), we have that 1 A[f]= lim g(⌧) g(0) =0, ⌧ + 7! 1 ⌧ provided that g only takes ﬁnite values; in particular, A[w˙ ]=0.

An averaging of f x = w˙ 2k yields the Virial Theorem: · A[f x]= 2 A[k] · Fluctuating forces

Langevin (1908) studies Brownian particles suspended in a viscous ﬂuid at rest by means of macroscopic Newtonian mechanics: mx¨ = f + F,

1 where to standard Stokesian drag f = µ x˙ he adds ﬂuctuating force F, giving the motion a stochastic character; • makes use of lower alternative: • 2 1 d 1 1 d m (x x)+ µ (x x)=mx˙ x˙ + F x; 2 dt2 · 2 dt · · · takes arithmetic mean (no probabilistic expectation!) over large • collection of identical particles; characterizes the ﬂuctuating force by assuming that the mean of the • virial term F x be null. · Particle systems

Set • d (m x˙ x ) m x˙ x˙ = f x , f = f + f e dt i i · i i i · i i · i i ij i j=i X X X X6 On choosing o g = mass center, get Virial Theorem in split • ⌘ form:

A[F g]= 2 A[K ] · g 8 e F := fi , 2Kg := Mg˙ g˙ ,M:= mi > · > > P P > <> A (x x ) f + x f e = 2 A[K ] i j · ij i · i rel i>j > ⇥ X X X ⇤ > 2Krel := mix˙ i x˙ i > · > :> P I.2 Deterministic Virial, Continuum and Eulerian Eulerian balance of linear momentum

ni 0 = d ni + divT ⇢v˙ , d = dist. force, T = Cauchy stress After dyadic multiplication by ' and integration by parts,

0 = ' d ni + ' c (grad ')T T ⇢ ' v˙ , c = Tn, ⌦ ⌦ ⌦ ZPt Z@Pt ZPt ZPt where c = contact forces at a point of @Pt. For ' x, ⌘ d ⇢ x v˙ = ⇢ x x˙ ⇢ v v , v = x˙ , P ⌦ dt P ⌦ P ⌦ Z t ⇣ Z t ⌘ Z t so that d x d ni + x c T T = ⇢ x x˙ ⇢ v v. P ⌦ @P ⌦ P dt P ⌦ P ⌦ Z t Z t Z t ⇣ Z t ⌘ Z t Recall f x = w˙ 2k A[f x]= 2 A[k] · ) · Now, by taking the trace of d x d ni + x c T T = ⇢ x x˙ ⇢ v v , P ⌦ @P ⌦ P dt P ⌦ P ⌦ Z t Z t Z t ⇣ Z t ⌘ Z t we get

ni x d + x c tr T = W˙ (Pt) 2K(Pt) Pt · @Pt · Pt 9 R R R 1 ) W (Pt)= ⇢x x˙ ,K(Pt):= ⇢ v v > Pt · 2 Pt ⌦ = R R ;> ni A x d + x c tr T = 2 A[K(Pt)], ) · · ZPt Z@Pt ZPt ⇥ ⇤ a statement of the Virial Theorem in Continuum Mechanics. By taking the symmetric part of d x d ni + x c T T = ⇢ x x˙ ⇢ v , P ⌦ @P ⌦ P dt P ⌦ P ⌦ Z t Z t Z t ⇣ Z t ⌘ Z t we get

ni ] 1 sym x d + x c symT = I¨(Pt), P ⌦ @P ⌦ P 2 ⇣ Z t Z t ⌘ Z t where

I (Pt):= ⇢ x x = Euler inertia tensor of part Pt, ⌦ ZPt T ] := ⇢ v v + T = virial stress. ⌦ Some call T ] the Reynolds stress, because of the form of momentum balance typical of ﬂuid mechanics that O. Reynolds liked:

ni @t(⇢v)+div(⇢ v v T)=d , ⌦ where ⇢ v v = convective current and T = diffusive current. ⌦ Lecture II - Probabilistic Virial

II.1 Recap of Statistical Mechanics – The microcanonical ensemble – Microcanonical and time averages – How to construct an equilibrium thermodynamics II.2 The Equipartition Theorem II.3 The Virial Theorem – The Virial Theorem and the gas law

Main references: - J.P. Sethna, Entropy, Order Parameters, and Complexity. Oxford Master Series in Physics, Oxford Press, 2006. - K. Huang, Statistical Mechanics. 2nd Ed. J. Wiley, 1987. II.1 Recap of Statistical Mechanics The microcanonical ensemble

Consider classical Hamiltonian system N identical particles of invariable mass m • confined in a 3D box of volume V (finite molecular volume V/N) • 1 1 2 H(q, p)=K(p)+U(q),K(p)= m p ,f(q)= @ U(q). • 2 | | q If both particle-particle and particle-wall collisions entail no energy losses, then the total energy of the system is conserved. Three macroscopic conservation conditions – of number, volume, and energy – define the Gibbsian microcanonical ensemble N,V,E . { } Note @ other motion-related conserved quantities: confinement in a box destroys both translational and rotational symmetries. for all practical purposes, when a system is in equilibrium, all mi- • crostates of the system having the same energy initially, and hence forever, are regarded as equivalent; all microstates accessed by the system along one of its trajectories • have equal energy; (ergodicity concept) all ly many microstates of a given energy • 1 are going to be visited, sooner or later, except perhaps a subset of measure zero .

We now proceed to measure (not to count!) ‘how many’ initial states there are for a given assignment of initial energy. ⌦(E)= ✓(E H(q, p))dqdp , qp-plane, ✓( )=Heaviside f. • Z ⌘ · ZZ = volume of subgraph of H(q, p)=E

⌦(E + dE) ⌦(E)= ✓(E + dE H(q, p)) ✓(E H(q, p)) dqdp • ZZ = volume of (inﬁnitesimal) energy shell Heaviside vs. Dirac

Let D denote distributional differentiation. The distributional rela- tionships between the Heaviside and Dirac functions is

f(x )= ✓(x x0)f 0(x) dx = D✓(x x )f(x)dx 0 0 ZIR ZIR for all test functions f. Hence, D✓(x x )=(x x ). 0 0 ⌦0(E)= (E H(q, p)) dqdp , ( )=Dirac f. • · ZZ volume partition f. := E ⌦(E)= ✓(E H(q, p)) dqdp • 7! ZZ surface partition f. := E !(E)= (E H(q, p)) dqdp • 7! ZZ measures boundary of region in where H(q, p) E. Z  Note that ⌦0(E)=!(E) 1 microcanonical probability meas.:= ⇢˜ (z)= (E H(z)) • E !(E) Exercise

Let h(x) 0 for x<0, h(x) >efor all x>x4. Show that ⌘ ⌦(e)= ✓(e h(x)) dx =(x x )+(x x ) • 2 1 4 3 ZIR !(e)= (e h(x)) dx =1+1+1+1, • ZIR so that integration of d!(e)=(e h(x)) dx literally counts all microstates x1,...,x4 where h(x)=e. Microcanonical and time averages

microcanonical average of F : • 1 F (E) := F (q, p) (E H(q, p)) dqdp h i !(E) ZZ (E,d)-trajectory: • a trajectory of the N-particle system under study, conﬁned within a volume V , of energy E and duration d that is, a solution t (q(t),p(t)),t [0,d) of the motion equa- 7! 2 tions which complies with conﬁnement condition and initial con- ⇥ ditions (q(0),p(0)), and is such that H(q(0),p(0)) = E. 1 d ⇤ time average of F : F (E) := lim F (q(t),p(t))dt. • d d !1 Z0 Microcanonical and time averages (cont.ed)

Given a (E,d)-trajectory, one assumes that (i) • 1 d F (E) F (q(t),p(t))dt , h i ' d Z0 with an approximation that becomes better and better the more accurately the given trajectory visits all parts of the state space where the energy takes the prescribed value; (ii) when d , the system tends to statistical equilibrium, that • !1 is, to a situation when all quantities that are not conserved (such as F itself) are anyway independent of the initial conditions. SM, MD, and the Ergodic Hypothesis

A basic assumption in SM – oftentimes referred to as the ergodic hypothesis – is that, for each given E, F (E) = F (E), E, h i 8 so that the microcanonical average can be calculated from a time average. The role of MD is precisely to allow for evaluating statistical averages via time averages along trajectories:

statistical average at equilibrium time average along trajectories ' SM MD

Note| Averages and{z trajectories are} to| be chosen consistent{z with the} ensemble that ﬁts the system under study. MD, CM, and experimental validation

Current MD simulations: basic cell of attomole size • X duration of nanosecond order • T At the macroscopic scale of CM, the space-time regions considered by MD are to be re- • X ⇥ T garded as (point, instant) pairs (x, t) the measured value F (x, t) of a ﬁeld F should conﬁrm the predic- • tion F (x, t) derived from solving an appropriate initial/boundary- value problem e To relate F (x, t) with F as evaluated by means of an MD simulation, basic cell should be “large enough” (i.e., much larger than the • correlation length of the spatial correlation function of interest) duration should be “long enough” (i.e., much longer than the re- • laxation time of the property of interest) Providing these quantitative conditions are met, MD furnishes a bridge between SM and CM: statistical average at equilibrium time average along trajectories ' SM MD measured value . | {z } | {z } ' CM | {z } How to construct an equilibrium thermodynamics

(when working with the microcanonical ensemble in the background) Step 1. Deﬁne an entropy function S = S(E,V,N)=k log Z(E)() B ⇤ [ volume entropy ifb Z(E)=⌦(E), surface entropy if Z(E)=!(E)] Step 2. For whatever entropy, solve for E the implicit equation ( ): ⇤ E = U(S, V, N)=internal-energy function

Step 3. Deﬁne temperatureb T := @SE Step 4. Deﬁne the Helmholtz free energy to be the negative of the Legen- dre transform of the internal energy function:

1 (V,N,T)=E T S(E,V,N),T = @ S. E b b II.2 The Equipartition Theorem

Consider (microcanonical) expected value 1 a @ H (E)= a @ H(z) (E H(z)) dz , h i aj i !(E) i aj ZZ where ai,aj = microscopic degrees of freedom of system whose Hamil- tonian is H = H(q, p).

Generalized Equipartitionb Theorem

(E)=(kBT ) ij ,T= equilibrium temperature

Interpretation measures ‘conjugation wrt expectation’ of metavelocity ai and metaforce @aj H; GEP is a statement of ‘or- thogonality wrt expectation’ of all (metavelocity, metaforce) pairs of different indices. Proof of GEP. 1/3

On recalling that (E H(z)) = D ✓(E H(z)), the integral ai @a H(z) (E H(z)) dz can be written as j ZZ a @ H(z) D✓(E H(z)) dz = @ a @ H(z) ✓(E H(z)) dz ; i aj E i aj ZZ ⇣ ZZ ⌘ moreover, a @ H(z)=a @ (H(z) E)=@ (H(z) E)a ) (H(z) E) . i aj i aj aj i ij Hence,

!(E) (E)= @ @ (E H(z))a ✓(E H(z)) dz i aj E aj i ZZ ⇣ ⌘ + @ (E H(z)) ✓(E H(z)) dz . E ij ⇣ ZZ ⌘ Proof of GEP. 2/3

The integral @E @a (E H(z))ai ✓(E H(z)) dz van- • j ZZ ishes, because ⇣ ⌘

@ (E H(z))a ✓(E H(z)) dz = @ (E H(z))a dz aj i aj i ZZ ZR = (E H(z))ai (n@ )j da@ =0, @ R R Z R for the region of where H(z)

(E H(z)) ✓(E H(z)) dz = (E H(z)) , whence • ZZ ZR @ (E H(z)) ✓(E H(z)) dz = vol( )=⌦(E). E R ⇣ ZZ ⌘ Proof of GEP. 3/3

Thus,

!(E) (E)= @ @ (E H(z))a ✓(E H(z)) dz i aj E aj i ZZ ⇣ ⌘ + @ (E H(z)) ✓(E H(z)) dz E ij ⇣ ZZ ⌘ = ⌦(E)ij. At this point, with the sequential use of

!(E)=⌦0(E) • 1 S = k ln Z(E),Z(E)=⌦(E); @ S = T • B E we deduce that

⌦(E) 1 kB (E)= ij = ij = ij ⌦0(E) (log ⌦(E))0 @ES(E,V,N) = kBT ij ⇤ b Equipartition of what? and among whom?

1 2 2 1 1 2 1 Consider an undamped harmonic oscillator: H(q, p)= m ! q + m p = (q @qH + p @pH); 2 2 2 set H(q, p)=E; compute the GEP expression for each microscopic DOF: (E)=kBT = (E) .

Then, the energy expectation is (E)=kBT, and is split 1 into equal parts kBT for each microscopic DOF. More generally, 2 the GEP yields:

(E)=kBT = (E)(i (unsummed)=1, 2,...n) . 1 n Whenever qi @q H + pi @p H = H(q, p), the total energy of the 2 i i i=1 system is splitX into as many equal parts as the microscopic DOFs. II.3 The Virial Theorem

On recalling the motion equations, n n n < q @ H>(E)= < q p˙ >(E)= < q f (q)>(E)=n (k T ) . i qi i i i i B i=1 i=1 i=1 X X X n The construct qi fi(q) is called the virial of the system of forces i=1 acting on the system,X in the conﬁguration speciﬁed by q. At statisti- n cal equilibrium, the function E < qi fi(q) >(E) delivers the 7! i=1 ensemble average of the virial. The relationX