1. 1. A tricritical point is a point in a phase diagram where a line of second-order critical points meets a line of first-order critical points. We can model this by consider a Landau-Ginsburg free energy density of the form fLG = am? + bm' + cm® where we must have c > 0 for stability but a and b can be negative as well as positive. (a) Show that a second-order transition occurs at a=0 when b>0. (b) Taking a>0 and b<0, sketch fLG at a point on the first-order line. Find the first-order line by solving fic(m) = 0 and fLG(m) = 0 simultaneously. (c) Where is the tricritical point in the (a, b) plane? Sketch the phase diagram in %3D %3D that plane, showing the 1st- and 2nd-order lines as well as the tricritcal point.

1. 1. A tricritical point is a point in a phase diagram where a line of second-order critical points meets a line of first-order critical points. We can model this by consider a Landau-Ginsburg free energy density of the form fLG = am? + bm' + cm® where we must have c > 0 for stability but a and b can be negative as well as positive. (a) Show that a second-order transition occurs at a=0 when b>0. (b) Taking a>0 and b<0, sketch fLG at a point on the first-order line. Find the first-order line by solving fic(m) = 0 and fLG(m) = 0 simultaneously. (c) Where is the tricritical point in the (a, b) plane? Sketch the phase diagram in %3D %3D that plane, showing the 1st- and 2nd-order lines as well as the tricritcal point.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Consider the autonomous DE . dx/dt = x^2 - 8x + 7 (a) Its stable critical point is x= unstable critical point is x= (b) Applying five iterations of the Forward Euler Method to the above DE, using a step size h(0.1) and initial condition x(0)=3, yields: x(0.1)approximately x1 = x(0.2)approximately x2 = x(0.3)approximately x3 = x(0.4)approximately x4= x(0.5)approximately x5 =
The first image displays question 2 (the question I need help with). The second image displays question 8 which has been solved. It has been provided as an example just in case.
3. Derive the material balance for a tracer in a steady state PFR with the following second order reaction (k = -0.05 The reactor has a L mg*min' velocity of 0.1 m/min, a length of 3 m, and a starting tracer concentration of 50 mg/L. What is the effluent tracer concentration in the PFR?
Suppose that the equation dr = kr(M – 1) – Er dt models the population of fish in a lake, where harvesting occurs at the rate of Ex fish per month (E a positive constant call harvesting effort). If 0 < E < kM, (a) If 0 < E < kM, show that the population is logistic. (b) What is the limiting population? kM (c) Show that the maximum sustainable harvesting effort is Emax 2 (d) If E 2 kM, show that r(t) → 0 as t → x, that is the lake is eventually fished out.
Please help!
The number N(t) of supermarkets throughout the country that are using a computerized checkout system is described by the initial-value problem dN = N(1 dt 0.0004N), N(0) = 1. - (a) Use the phase portrait concept of Section 2.1 to predict how many supermarkets are expected to adopt the new procedure over a long period of time. supermarkets By hand, sketch a solution curve of the given initial-value problem. N 2500 2500 1200 1000 2000 2000 800 1500 1500 600 1000 1000 400 500 500 200 10 15 20 10 15 20 10 15 20 N 1200 1000 800 600 400 200 t 20 10 15 (b) Solve the initial-value problem and then use a graphing utility to verify the solution curve in part (a). N(t) = How many supermarkets are expected to adopt the new technology when t = 10? (Round your answer to the nearest integer.) supermarkets
2. A tank is initially filled with 100 L of salt solution with a concentration of 0.12 kg/L. At time t = 0, a solution containing 0.2 kg/L of salt is poured into the tank at a rate of 3 L/min. Simultaneously, a drain is opened in the bottom of the tank allowing the solution to leave the tank at a rate of 4 L/min. The tank is well-stirred. Let x(t) represent the amount of salt (in kg) in the tank after t minutes. Find the differential equation and initial condition for which x(t) is a solution. DO NOT SOLVE THE DIFFERENTIAL EQUATION. (t)= S t x' (t) = , x(0) =
2,2 The TUT cafeteria at the eMalahleni campus is home to an industrial juice dispenser with a volume of 20I. Initially the dispenser is filled with a juice blend containing 10% mango and 90% peach juice. An orange-mango blend containing 40% orange and 60% mango is entering the dispenser at a rate of 4/ / hr and the well-stirred mixture leaves at a rate of 4// hr. 2.2.1 Use a differential equation to model the amount of mango juice, M(1), in the dispenser at any time t and solve it. (The orange-mango blend starts to enter att=0)
Part II Solve the initial value problems. Obtain the particular solutions of the following: a. xy'+ y =0, y(4)=6 b. у'-1+4у?, у(1) -0 %3D c. xy'= y+3x* cos² ( y/x), y(1)=0 d. y'cosh x= sin’ y , 1 y(0)= 1– y° dx – /1– x²dy = 0, y(0)= е. %3D dy f. - у - ху, dx y(-1)=-1 dy g. dx -x y(0) =2 y