Universität Stuttgart

Universität Stuttgart

Fakultät 8, Fachbereich Physik

Prof. Dr. Rudolf Hilfer Tillmann Kleiner

Advanced Statistical Physics, SS 2017 Sheet 1

Problem 1:

Let K ⊆ R

be a convex set and f : K → R a convex function. Define

K

:= {(x, z) ∈ R

: x ∈ K, z ∈ R , z ≥ f (x)}. (1) a) Show that K

is a convex subset of R

.

b) For a general function f : K → R show that f is convex on K if and only if K

is a convex subset of R

.

Problem 2:

Let I be an open interval in R and f : I → R a convex function. Prove the following statements:

a) For each a, b ∈ I, a ≤ b there exist K

, k

∈ R such that k

≤ f(x) ≤ K

for all a ≤ x ≤ b.

b) For each a, b ∈ I, a ≤ b there exists L

∈ R such that |f(x) − f(y)| ≤ L

|x − y|

for all a ≤ x, y ≤ b.

c) The function f is continuous.

Hint: To prove statement b), first choose c, d ∈ I such that c + ≤ a and b ≤ d − for some > 0. Let x, y ∈ [a, b] with x < y and define z = y + . Find λ such that y = λz + (1 − λ)x.

Use the inequality characterising the convexity of f and solve for f (y) − f(x).

Remark: Similar statements hold for convex functions defined on convex subsets of R

.

Problem 3:

Show that each of the following functions is convex or concave and calculate their Leg- endre transforms.

a) The function √

1 + x

defined on R .

b) The function cos(x) defined on (−π/2, π/2).

c) The function |x| defined on R .

Problem 4:

Consider the function U = U (S, V ) = S

e

for V ∈ R and S > 0.

a) Show that U is a convex function.

b) Define T =

. Calculate the partial Legendre transform F (T, V ) = U (S, V ) − T S of U obtained by replacing S with T .

c) Define P = −

. Calculate the Legendre transform G(T, P ) = U (S, V )−T S−P V

of U obtained by replacing S with T and V with P .