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Statistical Physics and Thermodynamics - SoSe 20 Freie Universität Berlin Monday May 25, 2020 Due date: Monday June 1, 2020 Please upload your solution as one pdf file on whiteboard. Unreadable solutions and files submitted after the due date will get no points. Prof. Dr. Roland R. Netz Tutors: Cihan Ayaz (0.3.06)
[email protected] Shane Carlson (0.3.34)
[email protected] Sina Zendehroud (0.3.33)
[email protected] Statistical Mechanics: Applications of Canonical Ensembles in Soft-Matter Physics 1 Lecture Related Preliminaries a) (2P) In the lecture, when deriving the p.d.f. for canonical ensembles, the following approximation was made: S(U − U1) ≈ S(U)− U1 ∂S ∂U , for U � U1. (1) In the approximation above, the term U21/2 · ∂2S/∂U2 was neglected since it scales like ∼ 1/N. Here, show that U21 ∂2S ∂U2 = − U 2 1 CV T 2 , (2) where CV denotes the heat capacity of the reservoir and explain shortly in your own words why CV T � U1 utilizing the concept of heat capacity. b) (3P) In the lecture, we showed that the energy fluctuations are given by〈 (U − 〈U〉)2 〉 = kBT 2CV . (3) Show in an analogous manner that the third centered moment can be computed from 〈 (U − 〈U〉)3 〉 = k2B ( T 4 ∂CV ∂T + 2T 3CV ) . (4) Hint: First, show that ∂2〈U〉/∂β2 = 〈 (U − 〈U〉)3 〉 . 2 DNA Zipping Consider a simple model for DNA unbinding: The unbinding of the double helix is like unzipping of a zipper; a base pair (bp) can open only if all bp’s to its left are already open (see figure above). The DNA has N bp’s, each of which can be in one of two states: An open state with energy 0 and a closed one with energy −ε. Hence, the total energy of the double helix is E = −nε, where n denotes the number of closed bp’s. The system is coupled to a heat reservoir. a) (3P) Show that the average number of closed bp’s 〈n〉 is given by 〈n〉 = 1 e−βε − 1 − N + 1 e−(N+1)βε − 1 . (5) b) (2P) Discuss the high temperature βε� 1 and low temperature limits βε� 1. http://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-netz/lehre/ Page 1 Statistical Physics and Thermodynamics - SoSe 20 Freie Universität Berlin 3 Freely Jointed Polymer Chain Under Tension In problem set 2, we considered a freely-jointed polymer chain in 2D, where each segment could only be oriented along the x or y direction. In this problem, we consider a freely-jointed chain in 3D that is coupled to a heat bath at temperature T . Each of the N segments has a length of l. The segments can rotate by arbitrary polar, 0 ≤ ϑ < π,="" and="" azimuthal,="" 0="" ≤="" ϕ="">< 2π, angles independently of each other. one end of the chain is fixed at the origin, on the other end at r, a constant force f pulls on the chain. the force is directed along the z-axis (see figure above). the hamiltonian of the chain under tension is given by h = −f · r, where f ‖ ẑ. (6) a) (2p) show that the partition function of the chain under tension is given by z = ( 4π sinh(βfl) βfl )n , (7) where f = |f | is the amplitude of the force. hint: the unit surface element in spherical coordinates is given by dω = dϑdϕ sinϑ. b) (3p) we are interested in the end-to-end distance χ along the z axis, i.e. χ = r · ẑ = n∑ n=1 l cosϑn. (8) show that the average end-to-end distance 〈χ〉 = x is given by x nl = coth(βfl)− 1 βfl . (9) hint: use a suitable derivative of the partition fucntion z. c) (2p) consider the high temperature βfl � 1 and low temperature βfl � 1 limits and express f as a function of x for both cases. hint: you can use coth(x) ≈ 1 for x� 1 and coth(x) ≈ 1/x + x/3 for x→ 0. d) (3p) from the so-called gibbs free energy g(t, f) = −kbt lnz, we can compute the system variables s(t, f) = − ∂g ∂t ∣∣∣∣ f x(t, f) = − ∂g ∂f ∣∣∣∣ t . (10) we now consider the limit βfl� 1, i.e. where f(x) is given by your result in c). use a suitable legendre transform to obtain the free energy f (t,x) as a function of the temperature and the average end-to-end distance along the z-axis and from this, compute s(t,x) in the limit βfl� 1. hint: go back to problem set 4 and have a look at question 1 e). http://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-netz/lehre/ page 2 2π,="" angles="" independently="" of="" each="" other.="" one="" end="" of="" the="" chain="" is="" fixed="" at="" the="" origin,="" on="" the="" other="" end="" at="" r,="" a="" constant="" force="" f="" pulls="" on="" the="" chain.="" the="" force="" is="" directed="" along="" the="" z-axis="" (see="" figure="" above).="" the="" hamiltonian="" of="" the="" chain="" under="" tension="" is="" given="" by="" h="−f" ·="" r,="" where="" f="" ‖="" ẑ.="" (6)="" a)="" (2p)="" show="" that="" the="" partition="" function="" of="" the="" chain="" under="" tension="" is="" given="" by="" z="(" 4π="" sinh(βfl)="" βfl="" )n="" ,="" (7)="" where="" f="|f" |="" is="" the="" amplitude="" of="" the="" force.="" hint:="" the="" unit="" surface="" element="" in="" spherical="" coordinates="" is="" given="" by="" dω="dϑdϕ" sinϑ.="" b)="" (3p)="" we="" are="" interested="" in="" the="" end-to-end="" distance="" χ="" along="" the="" z="" axis,="" i.e.="" χ="r" ·="" ẑ="N∑" n="1" l="" cosϑn.="" (8)="" show="" that="" the="" average="" end-to-end="" distance="" 〈χ〉="X" is="" given="" by="" x="" nl="coth(βfl)−" 1="" βfl="" .="" (9)="" hint:="" use="" a="" suitable="" derivative="" of="" the="" partition="" fucntion="" z.="" c)="" (2p)="" consider="" the="" high="" temperature="" βfl="" �="" 1="" and="" low="" temperature="" βfl="" �="" 1="" limits="" and="" express="" f="" as="" a="" function="" of="" x="" for="" both="" cases.="" hint:="" you="" can="" use="" coth(x)="" ≈="" 1="" for="" x�="" 1="" and="" coth(x)="" ≈="" 1/x="" +="" x/3="" for="" x→="" 0.="" d)="" (3p)="" from="" the="" so-called="" gibbs="" free="" energy="" g(t,="" f)="−kBT" lnz,="" we="" can="" compute="" the="" system="" variables="" s(t,="" f)="−" ∂g="" ∂t="" ∣∣∣∣="" f="" x(t,="" f)="−" ∂g="" ∂f="" ∣∣∣∣="" t="" .="" (10)="" we="" now="" consider="" the="" limit="" βfl�="" 1,="" i.e.="" where="" f(x)="" is="" given="" by="" your="" result="" in="" c).="" use="" a="" suitable="" legendre="" transform="" to="" obtain="" the="" free="" energy="" f="" (t,x)="" as="" a="" function="" of="" the="" temperature="" and="" the="" average="" end-to-end="" distance="" along="" the="" z-axis="" and="" from="" this,="" compute="" s(t,x)="" in="" the="" limit="" βfl�="" 1.="" hint:="" go="" back="" to="" problem="" set="" 4="" and="" have="" a="" look="" at="" question="" 1="" e).="" http://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-netz/lehre/="" page="">