Given the Ode Find the Constant Solutions and Its Stability

Systems of Differential Equations

Martha L. Abell , James P. Braselton , in Introductory Differential Equations (Fifth Edition), 2018

6.7 Nonlinear Systems

We now turn our attention to systems in which at least one of the equations is not linear. These nonlinear systems possess many of the same properties as linear systems. In fact, we rely on our study of the classification of equilibrium points of linear systems to assist us in our understanding of the behavior of nonlinear systems. For example, consider the system

(6.27) $\begin{array}{cc}\hfill dx/dt& =y\hfill \\ \hfill dy/dt& =x+x^2\hfill \end{array}$

which is nonlinear because of the $x^2$ term in the second equation. If we remove this nonlinear term, we obtain the linear system

(6.28) $\begin{array}{cc}\hfill dx/dt& =y\hfill \\ \hfill dy/dt& =x\hfill \end{array}.$

This system has an equilibrium point at $(0,0)$ , which is also an equilibrium point of the system of nonlinear equations. Using the techniques discussed in Section 6.6, we can quickly show that the linear system has a saddle point at $(0,0)$ . We show several trajectories of this system together with its direction field in Fig. 6.19A. Next, in Fig. 6.19B, we graph several trajectories of the nonlinear system along with its direction field. We see that the nonlinear term affects the behavior of the trajectories. However, the, behavior near $(0,0)$ is "saddle-like" in that solutions appear to approach the origin but eventually move away from it. When we zoom in on the origin in Fig. 6.19C, we see how much it resembles a saddle point. Notice that the curves that separate the trajectories resemble the lines $y=x$ and $y=-x$ , the asymptotes of trajectories in the linear system, near the origin. These curves, along with the origin, form the separatrix, because solutions within the portion of the separatrix to the left of the y-axis behave differently than solutions in other regions of the xy-plane. In this section, we discuss how we classify equilibrium points of nonlinear systems by using an associated linear system in much the same way as we analyzed systems (6.27) and (6.28).

Figure 6.19. (A) Trajectories of corresponding linear system with direction field. (B) Trajectories of nonlinear system with direction field. (C) Trajectories of nonlinear system with direction field near the origin.

The equilibrium solutions to a system of differential equations in which each differential equation does not explicitly depend on the independent variable (typically, t) are the constants solutions of the system. Thus, the equilibrium solutions of $x_1^{\prime }=f_1(x_1,x_2,\dots ,x_n)$ , $x_2^{\prime }=f_2(x_1,x_2,\dots ,x_n)$ , …, $x_n^{\prime }=f_n(x_1,x_2,\dots ,x_n)$ are found by solving the system $f_1(x_1,x_2,\dots ,x_n)=0$ , $f_2(x_1,x_2,\dots ,x_n)=0$ , …, $f_n(x_1,x_2,\dots ,x_n)=0$ for $x_1$ , $x_2$ , …, $x_n$ .

When working with nonlinear systems, we can often gain a great deal of information concerning the system by making a linear approximation near each equilibrium point of the nonlinear system and solving the linear system. Although the solution to the linearized system only approximates the solution to the nonlinear system, the general behavior of solutions to the nonlinear system near each equilibrium point is the same as that of the corresponding linear system in many cases. The first step toward approximating a nonlinear system near each equilibrium point is to find the equilibrium points of the system and to linearize the system at each of these points.

Recall from multivariable calculus that if $z=F(x,y)$ is a differentiable function, the tangent plane to the surface S given by the graph of $z=F(x,y)$ at the point $(x_0,y_0)$

(6.29) $\begin{array}{cc}\hfill z=& F_x(x_0,y_0)(x-x_0)+F_y(x_0,y_0)(y-y_0)\hfill \\ \hfill & +F(x_0,y_0)\hfill \end{array}$

Near each equilibrium point $(x_0,y_0)$ of the (nonlinear) autonomous system

(6.30) $\begin{array}{cc}\hfill dx/dt& =f(x,y)\hfill \\ \hfill dy/dt& =g(x,y)\hfill \end{array},$

System (6.30) is autonomous because $f(x,y)$ and $g(x,y)$ do not explicitly depend on the independent variable, which is t in this situation.

under certain conditions the system's solution(s) can be approximated with the system

(6.31) $\begin{array}{cc}\hfill dx/dt=& f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)\hfill \\ \hfill & +f(x_0,y_0)\hfill \\ \hfill dy/dt=& g_x(x_0,y_0)(x-x_0)+g_y(x_0,y_0)(y-y_0)\hfill \\ \hfill & +g(x_0,y_0),\hfill \end{array}$

where we have used the tangent plane to approximate $f(x,y)$ and $g(x,y)$ in the two-dimensional autonomous system (6.30). Because $f(x_0,y_0)=0$ and $g(x_0,y_0)=0$ (Why?), the approximate system is

(6.32) $\begin{array}{cc}\hfill dx/dt& =f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)\hfill \\ \hfill dy/dt& =g_x(x_0,y_0)(x-x_0)+g_y(x_0,y_0)(y-y_0)\hfill \end{array},$

which can be written in matrix form as

(6.33) ${\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)}^{\prime }=\left(\begin{array}{cc}\hfill f_x(x_0,y_0)\hfill & \hfill f_y(x_0,y_0)\hfill \\ \hfill g_x(x_0,y_0)\hfill & \hfill g_y(x_0,y_0)\hfill \end{array}\right)\left(\begin{array}{c}\hfill x-x_0\hfill \\ \hfill y-y_0\hfill \end{array}\right).$

Note that we often call system (6.33) the linearized system corresponding to the (nonlinear) system (6.30) or the associated linearized system due to the fact that we have removed the nonlinear terms from the original system.

If $f(x_0,y_0)=0$ and $g(x_0,y_0)=0$ , the equilibrium point $(x_0,y_0)$ of the system $\{\begin{array}{c}dx/dt=f(x,y)\hfill \\ dy/dt=g(x,y)\hfill \end{array}$ is classified by the eigenvalues of the matrix

(6.34) $\mathbf{J}(x_0,y_0)=\left(\begin{array}{cc}\hfill f_x(x_0,y_0)\hfill & \hfill f_y(x_0,y_0)\hfill \\ \hfill g_x(x_0,y_0)\hfill & \hfill g_y(x_0,y_0)\hfill \end{array}\right),$

which is called the Jacobian matrix. After determining the Jacobian matrix for each equilibrium point, we find the eigenvalues of the matrix in order to classify the corresponding equilibrium point according to the following criteria.

Notice that the linearization must be carried out for each equilibrium point.

6.7.0.1 Classification of Equilibrium Points of a Nonlinear System

Let $(x_0,y_0)$ be an equilibrium point of system (6.30) and let ${\lambda }_1$ and ${\lambda }_2$ be eigenvalues of the Jacobian matrix (6.34) of the associated linearized system about the equilibrium point $(x_0,y_0)$ .

1.

If $(x_0,y_0)$ is classified as an asymptotically stable or unstable improper node (because the eigenvalues of $\mathbf{J}(x_0,y_0)$ , are real and distinct), a saddle point, or an asymptotically stable or unstable spiral in the associated linear system, $(x_0,y_0)$ has the same classification in the nonlinear system.

2.

If $(x_0,y_0)$ is classified as a center in the associated linear system, $(x_0,y_0)$ may be a center, unstable spiral point, or asymptotically stable spiral point in the nonlinear system, so we cannot classify $(x_0,y_0)$ in this situation (see Exercise 28).

3.

If the eigenvalues of $\mathbf{J}(x_0,y_0)$ are real and equal, then $(x_0,y_0)$ may be a node or a spiral point in the nonlinear system. If ${\lambda }_1⩽{\lambda }_2<0$ , then $(x_0,y_0)$ is asymptotically stable. If ${\lambda }_1⩽{\lambda }_2>0$ , then $(x_0,y_0)$ is unstable.

These findings are summarized in Table 6.2.

Table 6.2. Classification of Equilibrium Point in Nonlinear System

Eigenvalues of J( x 0 , y 0 )	Geometry	Stability
λ 1,λ 2 real; λ 1 >λ 2 > 0	Improper node	Unstable
λ 1,λ 2 real; λ 1 =λ 2 > 0	Node or spiral point	Unstable
λ 1,λ 2 real; λ 2 <λ 1 < 0	Improper node	Asymptotically stable
λ 1,λ 2 real; λ 1 =λ 2 < 0	Node or spiral point	Asymptotically stable
λ 1,λ 2 real; λ 2 < 0 <λ 1	Saddle point	Unstable
λ 1 =α +βi, λ 2 =α −βi, β ≠ 0, α > 0	Spiral point	Unstable
λ 1 =α +βi, λ 2 =α −βi, β ≠ 0, α < 0	Spiral point	Asymptotically stable
λ 1 =βi, λ 2 = −βi, β ≠ 0	Center or spiral point	Inconclusive

Example 6.39

Find and classify the equilibrium points of $\{\begin{array}{c}{x}^{\prime }=1-y\hfill \\ y^{\prime }=x^2-y^2\hfill \end{array}$ .

Solution: We begin by finding the equilibrium points of this nonlinear system by solving $\{\begin{array}{c}1-y=0\hfill \\ x^2-y^2=0\hfill \end{array}$ . Because $y=1$ from the first equation, substitution into the second equation yields $x^2-1=0$ . Therefore, $x=\pm 1$ , so the two equilibrium points are $(1,1)$ and $(-1,1)$ . Because $f(x,y)=1-y$ and $g(x,y)=x^2-y^2$ , $f_x(x,y)=0$ , $f_y(x,y)=-1$ , $g_x(x,y)=2x$ , and $g_y(x,y)=-2y$ , so the Jacobian matrix is $\mathbf{J}(x,y)=\left(\begin{array}{cc}\hfill 0\hfill & \hfill -1\hfill \\ \hfill 2x\hfill & \hfill -2y\hfill \end{array}\right)$ . Next, we classify each equilibrium point by finding the eigenvalues of the Jacobian matrix of each linearized system.

For $(1,1)$ , we, obtain the Jacobian matrix $\mathbf{J}(1,1)=\left(\begin{array}{cc}\hfill 0\hfill & \hfill -1\hfill \\ \hfill 2\hfill & \hfill -2\hfill \end{array}\right)$ with eigenvalues that satisfy $\left|\begin{array}{cc}\hfill -\lambda \hfill & \hfill -1\hfill \\ \hfill 2\hfill & \hfill -2-\lambda \hfill \end{array}\right|={\lambda }^2+2\lambda +2=0$ . Hence, ${\lambda }_{1,2}=-1\pm i$ . Because these eigenvalues are complex-valued with negative real part, we classify $(1,1)$ as an asymptotically stable spiral in the associated linearized system. Therefore, $(1,1)$ is an asymptotically stable spiral in the nonlinear system.

For $(-1,1)$ , we obtain $\mathbf{J}(-1,1)=\left(\begin{array}{cc}\hfill 0\hfill & \hfill -1\hfill \\ \hfill -2\hfill & \hfill -2\hfill \end{array}\right)$ . In this case, the eigenvalues are solutions of $\left|\begin{array}{cc}\hfill -\lambda \hfill & \hfill -1\hfill \\ \hfill -2\hfill & \hfill -2-\lambda \hfill \end{array}\right|={\lambda }^2+2\lambda -2=0$ . Thus, ${\lambda }_1=\frac{1}{2}(-2+2\sqrt{3})=-1+\sqrt{3}>0$ and ${\lambda }_2=\frac{1}{2}(-2-2\sqrt{3})=-1-\sqrt{3}<0$ so $(-1,1)$ is a saddle point in the associated linearized system and this classification carries over to the nonlinear system. In Fig. 6.20A, we graph solutions to this nonlinear system approximated with the use of a computer algebra system. We can see how the solutions move toward and away from the equilibrium points by observing the arrows on the vectors in the direction field. □

Figure 6.20. (A) (1,1) is a stable spiral and (−1,1) is a saddle. (B) (0,0) is an unstable node, (0,5) is an asymptotically stable improper node, (7,0) is an asymptotically stable improper node, and (3,2) is a saddle point.

Note: In Example 6.39, the linear approximation about $(1,1)$ is ${\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)}^{\prime }=\left(\begin{array}{cc}\hfill 0\hfill & \hfill -1\hfill \\ \hfill 2\hfill & \hfill -2\hfill \end{array}\right)\left(\begin{array}{c}\hfill x-1\hfill \\ \hfill y-1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -y+1\hfill \\ \hfill 2x-2y\hfill \end{array}\right)$ or $\{\begin{array}{c}{x}^{\prime }=-y+1\hfill \\ y^{\prime }=2x-2y\hfill \end{array}$ . Notice that this (nonhomogeneous) linear system has an equilibrium point at $(1,1)$ . With the change of variable $u=x-1$ and $v=y-1$ , where the change of variable for the equilibrium point $(x_0,y_0)$ is $u=x-x_0$ and $v=y-y_0$ , we can transform this system to $\{\begin{array}{c}{u}^{\prime }=-v\hfill \\ v^{\prime }=2u-2v\hfill \end{array}$ with equilibrium point $(0,0)$ . Finding the eigenvalues of the matrix of coefficients indicates that $(0,0)$ is a saddle point, and we can sketch the phase portrait of the linearized system by translating the axes back to the original variables.

Example 6.40

Find and classify the equilibrium points of $\{\begin{array}{c}{x}^{\prime }=x(7-x-2y)\hfill \\ y^{\prime }=y(5-x-y)\hfill \end{array}$ .

Solution: The equilibrium points of this system satisfy $\{\begin{array}{c}x(7-x-2y)=0\hfill \\ y(5-x-y)=0\hfill \end{array}$ . Thus, $\{x=0\text{or}7-x-2y=0\}$ and $\{y=0\text{or}5-x-y=0\}$ . If $x=0$ , then $y(5-y)=0$ so $y=0$ or $y=5$ , and we obtain the equilibrium points $(0,0)$ and $(0,5)$ . If $y=0$ , then $x(7-x)=0$ , which indicates that $x=0$ or $x=7$ . The corresponding equilibrium points are $(0,0)$ (which we found earlier) and $(7,0)$ . The other possibility that leads to an equilibrium point is the solution to $\{\begin{array}{c}7-x-2y=0\hfill \\ 5-x-y=0\hfill \end{array}$ , which is $x=3$ and $y=2$ , resulting in the equilibrium point $(3,2)$ .

The Jacobian matrix is $\mathbf{J}(x,y)=\left(\begin{array}{cc}\hfill 7-2x-2y\hfill & \hfill -2x\hfill \\ \hfill -y\hfill & \hfill 5-x-2y\hfill \end{array}\right)$ . We classify each of the equilibrium points $(x_0,y_0)$ of the associated linearized system using the eigenvalues of $\mathbf{J}(x_0,y_0)$ :

$\mathbf{J}(0,0)=\left(\begin{array}{cc}\hfill 7\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 5\hfill \end{array}\right)$ ; ${\lambda }_1=7$ , ${\lambda }_2=5$ ; $(0,0)$ is an unstable node.

$\mathbf{J}(0,5)=\left(\begin{array}{cc}\hfill -3\hfill & \hfill 0\hfill \\ \hfill -5\hfill & \hfill -5\hfill \end{array}\right)$ ; ${\lambda }_1=-3$ , ${\lambda }_2=-5$ ; $(0,5)$ is an asymptotically stable improper node.

$\mathbf{J}(7,0)=\left(\begin{array}{cc}\hfill -7\hfill & \hfill -14\hfill \\ \hfill 0\hfill & \hfill -2\hfill \end{array}\right)$ ; ${\lambda }_1=-2$ , ${\lambda }_2=-7$ ; $(7,0)$ is an asymptotically stable improper node.

$\mathbf{J}(3,2)=\left(\begin{array}{cc}\hfill -3\hfill & \hfill -6\hfill \\ \hfill -2\hfill & \hfill -2\hfill \end{array}\right)$ ; ${\lambda }_1=1$ , ${\lambda }_2=-6$ ; $(3,2)$ is a saddle point.

In each case, the classification carries over to the nonlinear system. In Fig. 6.20B, we graph several approximate solutions and the direction field to this nonlinear system through the use of a computer algebra system. Notice the behavior near each equilibrium point.  □

Example 6.41

Investigate the stability of the equilibrium point $(0,0)$ of the nonlinear system

$\begin{array}{cc}\hfill \frac{dx}{dt}& =y+\frac{1}{2}x(x^2+y^2)\hfill \\ \hfill \frac{dy}{dt}& =-x+\frac{1}{2}y(x^2+y^2)\hfill \end{array}.$

Solution: First, we find the Jacobian matrix, $\mathbf{J}(x,y)=\left(\begin{array}{cc}\hfill \frac{3}{2}{x}^2+\frac{1}{2}{y}^2\hfill & \hfill 1+xy\hfill \\ \hfill -1+xy\hfill & \hfill \frac{1}{2}{x}^2+\frac{3}{2}{y}^2\hfill \end{array}\right)$ . Then, at the equilibrium point $(0,0)$ , we have $\mathbf{J}(0,0)=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill \end{array}\right)$ , so the linear approximation is

$dx/dt=ydy/dt=-x$

with eigenvalues ${\lambda }_{1,2}=\pm i$ . Therefore, $(0,0)$ is a (stable) center in the linearized system. However, when we graph the direction field for the original (nonlinear) system in Fig. 6.21A, we observe that $(0,0)$ is not a center. Instead, trajectories appear to spiral away from $(0,0)$ (see Fig. 6.21B), so $(0,0)$ is an unstable spiral point of the nonlinear system. The nonlinear terms not only affect the classification of the equilibrium point but also change the stability. Note: This is the only case in which we cannot assign the same classification to the equilibrium point in the nonlinear system as we do in the associated linear system. When an equilibrium point is a center in the associated linear system, then we cannot draw any conclusions concerning its classification in the nonlinear system.  □

Figure 6.21. (A) The direction field indicates that (0,0) is unstable. (B) All (nontrivial) trajectories spiral away from the origin. (C) Phase portrait.

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First-Order Equations

Martha L. Abell , James P. Braselton , in Introductory Differential Equations (Fourth Edition), 2014

Equilibrium Solutions of dy/dt = f(y)

Malthus's equation, dy/dt = ky is a member of a special category of separable equations because it has the form y′ = dy/dt = f(y). Such equations are called autonomous equations because the independent variable t does not appear in the equation on the right. We find that, although we are able to solve many equations of this form, we can learn much about the behavior of solutions without actually solving the differential equation.

An equilibrium solution of the autonomous first-order equation dy/dt = f(y) is a constant solution of the ODE: a function y(t) = C, C constant for which f(y) = 0. Consider the equation

$\frac{\text{d}y}{\text{d}t}=2y-y^2\text{or}\frac{1}{2y-y^2}\text{d}y=\text{d}t.$

(Note that this is a nonlinear ODE because it involves the term y ². We learn more about the differences in linear and nonlinear equations throughout this chapter.) Instead of solving by separating variables, let us use a graphical approach to determining properties of solutions. First, we locate the equilibrium solutions of the equation by solving 2y − y ² = 0 or y(2 − y) = 0. The roots of this equation are y = 0 and y = 2, so the equilibrium solutions are y(t) = 0 and y(t) = 2. Notice that each of these functions satisfies dy/dt = 2y − y ² because dy/dt = 0 and y(2 − y) = 0.

Next, we investigate the behavior of solutions on the intervals y < 0, 0 < y < 2, and y > 2 by sketching the phase line. After marking the two equilibrium solutions on the vertical line in Figure 2.8, we determine the sign of dy/dt on each of the three intervals listed above. This can be done by substituting a value of y on each interval into f(y)v2y − y ². For example, f(−1) = −3, so dy/dt < 0 if y < 0. We use an arrow directed downward to indicate that solutions decrease on this interval. In a similar manner, we find that f(1) = 1 and f(3) = − 3, so dy/dt > 0 if 0 < y < 2, and dy/dt < 0 if y > 2. We include arrows on the phase line directed upward and downward, respectively, to indicate this behavior. Based on the orientation, solutions that satisfy y(0) = a, where either 0 < a < 2 or a > 2 approaches y = 2 as $t\to \infty$ , so we classify y = 2 as asymptotically stable. Solutions that satisfy y(0) = a where a < 0 or 0 < a < 2 move away from y = 0 as $t\to \infty$ . We say that y = 0 is unstable because there are solutions that begin near y = 0 and move away from y = 0 as $t\to \infty$ . In Figure 2.9, we graph several solutions to this equation (in the ty-plane) along with the phase portrait to illustrate this behavior. Notice that solutions with initial value near y = 0 move away from this line while solutions that begin on 0 < y < 2 or y > 2 move toward y = 2 as $t\to \infty$ .

Figure 2.8. Phase line for dy/dt = 2y − y ².

Figure 2.9. Several solutions of dy/dt = 2y − y ².

Note: Sometimes, we refer to equilibrium solutions as steady-state solutions, because over time, the solution approaches a constant value that no longer depends on time t. For example, if we interpret the solution y(t) of the differential equation to represent the size of a population, then if y(0) = a where either 0 < a < 2 or a > 2, we expect the population to approach 2 as $t\to \infty$ .

Notice that the equation dy/dt = f(y) is always separable because it can be written in the form (1/f(y))dy = dt.

Example 2.2.4

Solve dy/dt = 2y − y ².

Solution

After separating variables, we use partial fractions

$\begin{array}{r}\frac{1}{2y-y^2}\text{d}y=\text{d}t,\\ \frac{1}{2}\left(\frac{1}{y}-\frac{1}{y-2}\right)\text{d}y=\text{d}t.\end{array}$

Now we integrate and apply properties of the logarithm to simplify

$\begin{array}{r}\frac{1}{2}\left(ln\left|y\right|-ln|y-2|\right)=t+C_1,\\ ln\left|\frac{y}{y-2}\right|=2t+C_2,\\ \frac{y}{y-2}=C_3{e}^{2t}.\end{array}$

Solving for y gives us

$y=\frac{2C_3{e}^{2t}}{C_3{e}^{2t}-1}=\frac{2}{1+C{e}^{-2t}},$

where C is arbitrary.

Note that from the general solution, we can obtain the equilibrium solution y = 2 by setting C = 0. However, there is no value of C for which this general solution gives us the solution y = 0. Thus, y = 0 is a singular solution for this equation. Combining these two solutions together means that all solutions of the equation dy/dt = 2y − y ² can be written as

$y=\frac{2}{1+C{e}^{-2t}}\text{or}y=0,$

where C is constant.

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Radiation, Atmospheric

Knut Stamnes , Gary E. Thomas , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

V.B Radiative-Convective Equilibrium

In a real, optically thick planetary atmosphere, the radiative equilibrium solution yields an unstable temperature gradient. When this gradient is surpassed, the atmosphere responds spontaneously to maintain a lapse rate equal to that given by adiabatic equilibrium. This value varies with the water vapor amount present, because of phase changes that occur in rising and falling air parcels. The average adiabatic lapse rate for the Earth is Γ   =   −6.5   K/km. An artifice that has been frequently used in the past is to assume that in regions where $\text{d}{\text{T}}_{\text{r}\text{e}}/\text{d}\text{z}<\Gamma$ (i.e., has larger negative values) the atmosphere spontaneously adjusts to maintain the constant lapse rate. This correction acts to cool the lower optically thick regions and also largely eliminates the interface discontinuity. However, this simple adjustment does not conserve energy at the intersection of the two solutions, because the upward irradiance issuing from the lower region is lower than in RE. It is necessary to move the intersection point upwards until the upward irradiances match. The height of this transition region, which we identify as the tropopause in our conceptual model, is located at heights beween 7 and 11   km (see Fig. 5), depending upon the optical depth.

In our more general radiative-convective solution, it is once again necessary to tune the optical depth to match the observations. A larger opacity is needed because of the greater efficiency of upward heat transport. Figure 5 shows that τ^*  =   4 provides a very good fit to the empirical 1976   model at all heights in the troposphere.

An algebraic expression for the radiative-convective solution T _rc is also readily obtained. We invoke once again the concept of the emission height, the level of maximum radiative cooling. By requiring that the temperature at this height (denoted z _e ) be equal to the effective temperature, T _e , it is trivial to find the expression for the surface temperature:

(18) ${\text{T}}_{\text{s}}={\text{T}}_{\text{e}}+|\Gamma |{\text{z}}_{\text{e}}={\text{T}}_{\text{e}}+|\Gamma |{\text{H}}_{\text{a}}\text{ln}(\tau */{\tau }_{\text{e}})$

where τ_e is the optical depth at the emission height z _e . As discussed previously, ${\tau }_{\text{e}}=\bar{\mu }$ . We may therefore combine the convective solution in the lower region and the RE solution in the upper region so that:

(19) $\begin{array}{l}\begin{array}{ll}{\text{T}}_{\text{r}\text{c}}={\text{T}}_{\text{e}}+|\Gamma |[{\text{H}}_{\text{a}}ln(\tau */\bar{\mu })-\text{z}]\hfill & (\text{z}\le {\text{z}}_{\text{t}})\hfill \end{array}\hfill \end{array}$

(20) $\begin{array}{l}\begin{array}{ll}{\text{T}}_{\text{r}\text{c}}={\text{T}}_{\text{e}}{(2)}^{-1/4}\hfill & (\text{z}>{\text{z}}_{\text{t}})\hfill \end{array}\hfill \end{array}$

where the tropopause height z _t is given by ${\text{z}}_{\text{t}}={\text{z}}_{\text{e}}+0.159\times {\text{T}}_{\text{e}}/|\Gamma |$ . Note that because the temperature of the transition region is very close to the skin temperature, we have ignored this small difference in Eq. (20).

We have obtained a realistic solution for the mean state of the atmosphere, in terms of the optical depth, the adiabatic lapse rate, and the scale height of the absorber. It is instructive to consider how the solution varies as the above parameters vary. For example, in the tropics the atmosphere is more humid, so τ^* is greater than average. Our solution predicts that the surface is hotter for a more opaque atmosphere. It also predicts that the tropical tropopause is higher than average. Both of these predictions agree with observations. However, the solution predicts the same tropopause temperature everywhere on the Earth, whereas it is well known that the tropical tropopause is both higher and cooler than at other latitudes. At this point, the failure of the simple one-dimensional model is to be expected, because it ignores horizontal transport of heat. In fact, the tropics is a net source of energy, whereas the polar regions are a net sink, as a result of horizontal transport by both the atmosphere and the ocean.

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Gravitational N-Body Problem (Classical)

D.C. Heggie , in Encyclopedia of Mathematical Physics, 2006

Collisional Evolution

Consider an isolated N -body system, which is supposed initially to be given by a spherically symmetric equilibrium solution of eqns [6] and [7], such as eqn [8]. The temperature decreases with increasing radius, and a gravothermal runaway causes the "collapse" of the core, which reaches extremely high density in finite time. (This collapse takes place on the long two-body relaxation timescale, and so it is not the rapid collapse, on a free-fall timescale, which the name rather suggests.)

At sufficiently high densities, the timescale of three-body reactions becomes competitive. These create bound pairs, the excess energy being removed by a third body. From the point of view of the one-particle distribution function, f, these reactions are exothermic, causing an expansion and cooling of the high-density central regions. This temperature inversion drives the gravothermal runaway in reverse, and the core expands, until contact with the cool envelope of the system restores a normal temperature profile. Core collapse resumes once more, and leads to a chaotic sequence of expansions and contractions, called gravothermal oscillations ( Figure 4 ).

Figure 4. Gravothermal oscillations in an N-body system with N = 65536. The central density is plotted as a function of time in units such that $t_{\text{cr}}=2\sqrt{2}$ . (Source: Baumgardt H, Hut P, and Makino J, with permission.)

The monotonic addition of energy during the collapsed phases causes a secular expansion of the system, and a general increase in all timescales. In each relaxation time, a small fraction of the masses escape, and eventually (it is thought) the system consists of a dispersing collection of mutually unbound single masses, binaries, and (presumably) stable higher-order systems.

It is very remarkable that the long-term fate of the largest self-gravitating N-body system appears to be intimately linked with the three-body problem.

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Stability of Flows

S. Friedlander , in Encyclopedia of Mathematical Physics, 2006

The Euler Equation: Linear andNonlinear Stability/Instability

We conclude this brief article with some discussion of instabilities in the inviscid Euler equations whose existence is likely to be important as a "trigger" for the development of instabilities in high-Reynolds-number viscous flows. As we mentioned, the Euler equations are very different from the Navier–Stokes equations in their mathematical structure. The Euler equations are degenerate and nonelliptic. As such, the spectrum of the linearized operator L _E is not amenable to standard spectral theory of elliptic operators. For example, unlike the Navier–Stokes operator, the spectrum of L _E is not purely discrete even in bounded domains. To define L _E we consider a steady Euler flow $\left\{{\mathit{U}}_0\left(\mathit{x}\right),P_0\left(\mathit{x}\right)\right\}$ , where

[12a] ${\mathit{U}}_0\cdot \nabla {\mathit{U}}_0=-\nabla P_0$

[12b] $\nabla \cdot {\mathit{U}}_0=0$

We assume that ${\mathit{U}}_0\in C^{\infty }$ . For the Euler equations, appropriate boundary conditions include zero normal component of ${\mathit{U}}_0$ on a rigid boundary, or periodicity conditions (i.e., flow on a torus) or suitable decay at infinity in an unbounded domain. The theorems that we will be describing have been proved mainly in the cases of the second and third conditions stated above. There are many classes of vector fields ${\mathit{U}}_0\left(\mathit{x}\right)$ , in two and three dimensions, that satisfy [12a] and [12b]. We write [4a] and [4b] in perturbation form as

[13] $\mathit{q}\left(\mathit{x},t\right)={\mathit{U}}_0\left(\mathit{x}\right)+\mathit{u}\left(\mathit{x},t\right)$

with

[14a] $\frac{\partial \mathit{u}}{\partial t}=L_{\text{E}}\mathit{u}+N\left(\mathit{u},\mathit{u}\right)$

[14b] $\nabla \cdot \mathit{u}=0$

Here

[15] $L_{\text{E}}\mathit{u}\equiv -({\mathit{U}}_0\cdot \nabla )\mathit{u}-(\mathit{u}\cdot \nabla ){\mathit{U}}_0-\nabla P_1$

[16] $N\left(\mathit{u},\mathit{u}\right)\equiv -(\mathit{u}\cdot \nabla )\mathit{u}-\nabla P_2$

Linear (spectral) instability of a steady Euler flow ${\mathit{U}}_0\left(\mathit{x}\right)$ concerns the structure of the spectrum of L _E. Assuming ${\mathit{U}}_0\in C^{\infty }\left(T^n\right)$ , the linear equation

[17] $\frac{\partial \mathit{u}}{\partial t}=L_{\text{E}}\mathit{u},\nabla \cdot \mathit{u}=0$

defines a strongly continuous group in every Sobolev space $W^{s,p}$ with generator L _E . We denote this group by $exp\left\{L_{\text{E}}t\right\}$ . For the issue of spectral instability of the Euler equation it proves useful to study not only the spectrum of L _E but also the spectrum of the evolution operator $\text{exp}\left\{L_{\text{E}}t\right\}$ . This permits the development of an explicit formula for the growth rate of a small perturbation due to the essential (or continuous) spectrum. It was proved by Vishik (1996) that a quantity Λ, refered to as a "fluid Lyapunov exponent" gives the maximum growth rate of the essential spectrum of $\text{exp}\left\{L_{\text{E}}t\right\}$ . This quantity is obtained by computing the exponential growth rate of a certain vector that satisfies a specific system of ODEs over the trajectories of the flow ${\mathit{U}}_0\left(\mathit{x}\right)$ . This proves to be an effective mechanism for detecting instabilities in the essential spectrum which result due to high-spatial-frequency perturbations. For example, for this reason any flow ${\mathit{U}}_0\left(\mathit{x}\right)$ with a hyperbolic fixed point is linearly unstable with growth in the sense of the L ₂-norm. In two dimensions, Λ is equal to the maximal classical Lyapunov exponent (i.e., the exponential growth of a tangent vector over the ODE $\stackrel{.}{x}=U_0(x)$ ). In three dimensions, the existence of a nonzero classical Lyapunov exponent implies that Λ>0. However, in three dimensions there are also examples where the classical Lyapunov exponent is zero and yet Λ>0. We note that the delicate issue of the unstable essential spectrum is strongly dependent on the function space for the perturbations and that Λ, for a given U ₀, will vary with this function space. More details and examples of instabilities in the essential spectrum can be found in references in the bibliography.

In contrast with instabilities in the essential spectrum, the existence of discrete unstable eigenvalues is independent of the norm in which growth is measured. From this point of view, such instabilities can be considered as "strong." However, for most flows ${\mathit{U}}_0\left(\mathit{x}\right)$ we do not know the existence of such unstable eigenvalues. For fully 3D flows there are no examples, to our knowledge, where such unstable eigenvalues have been proved to exist for flows with standard metrics. The case that has received the most attention in the literature is the "relatively simple" case of plane parallel shear flow. The eigenvalue problem is governed by the Rayleigh equation (which is the inviscid version of the Orr–Sommerfeld equation [11]):

[18] $\begin{array}{l}\left(U-\frac{\text{i}\lambda }{k}\right)\left[\frac{{\text{d}}^2}{\text{d}{z}^2}-k^2\right]\varphi -U″\varphi =0\\ \begin{array}{ccc}\varphi =0& \text{at}& z=\pm 1\end{array}\end{array}$

The celebrated Rayleigh stability criterion says that a sufficient condition for the eigenvalues λ to be pure imaginary is the absence of an inflection point in the shear profile $U\left(z\right)$ . It is more difficult to prove the converse; however, there have been several recent results that show that oscillating profiles indeed produce unstable eigenvalues. For example, if $U\left(z\right)=\text{sin}\text{<mi mathvariant="italic" is="true">mz</mi>}$ the continued fraction proof of Meshalkin and Sinai can be adapted to exhibit the full unstable spectrum for [18]. We note the "fluid Lyapunov exponent" Λ is zero for all shear flows; thus the only way the unstable spectrum can be nonempty for shear flows is via discrete unstable eigenvalues.

As we have discussed, it is possible to show that many classes of steady Euler flows are linearly unstable, either due to a nonempty unstable essential spectrum (i.e., cases where Λ>0) or due to unstable eigenvalues or possibly for both reasons. It is natural to ask what this means about the stability/instability of the full nonlinear Euler equations [14]–[16]. The issue of nonlinear stability is complex and there are several natural precise definitions of nonlinear stability and its converse instability.

One definition is to consider nonlinear stability in the energy norm L ² and the enstrophy norm H ¹, which are natural function spaces to measure growth of disturbances but are not "correct" spaces for the Euler equations in terms of proven properties of existence and uniqueness of solutions to the nonlinear equation. Falling under this definition is the most frequently employed method to prove nonlinear stability, which is an elegant technique developed by Arnol'd (cf. Arnol'd and Khesin (1998) and references therein). This is based on the existence of the so-called energy-Casimirs. The vorticity curl q is transported by the motion of the fluid so that at time t it is obtained from the vorticity at time t=0 by a volume-preserving diffeomorphism. In the terminology of Arnol'd, the velocity fields obtained in this manner at any two times are called isovortical. For a given field ${\mathit{U}}_0\left(\mathit{x}\right)$ , the class of isovortical fields is an infinite-dimensional manifold M, which is the orbit of the group of volume-preserving diffeomorphisms in the space of divergence-free vector fields. The steady flows are exactly the critical points of the energy functional E restricted to M. If a critical point is a strict local maximum or minimum of E, then the steady flow is nonlinearly stable in the space J ₁ of divergence-free vectors $\mathit{u}\left(\mathit{x},t\right)$ (satisfying the boundary conditions) that have finite norm,

[19] ${\Vert \mathit{u}\Vert }_{J_1}\equiv {\Vert \mathit{u}\Vert }_{L_2}+{\Vert \text{curl}\mathit{u}\Vert }_{L^2}$

This theory can be applied, for example, to show that any shear flow with no inflection points in the profile $U\left(z\right)$ is nonlinearly unstable in the function space J ₁, that is, the classical Rayleigh criterion implies not only spectral stability but also nonlinear stability.

We note that Arnol'd's stability method cannot be applied to the Euler equations in three dimensions because the second variation of the energy defined on the tangent space to M is never definite at a critical point ${\mathit{U}}_0\left(\mathit{x}\right)$ . This result is suggestive, but does not prove, that most Euler flows in three dimensions are nonlinearly unstable in the Arnol'd sense. To quote Arnol'd, in the context of the Euler equations "there appear to be an infinitely great number of unstable configurations."

In recent years, there have been a number of results concerning nonlinear instability for the Euler equation. Most of these results prove nonlinear instability under certain assumptions on the structure of the spectrum of the linearized Euler operator. To date, none of the approaches prove the definitive result that in general linear instability implies nonlinear instability. As we have remarked, this is a much more delicate issue for Euler than for Navier–Stokes because of the existence, for a generic Euler flow, of a nonempty essential unstable spectrum. To give a flavor of the mathematical treatment of nonlinear instability for the Euler equations, we present one recent result and refer the interested reader to articles listed in the "Further reading" section for further results and discussions.

In the context of Euler equations in two dimensions, we adopt the following definition of Lyapunov stability.

Definition 4

An equilibrium solution ${\mathit{U}}_0\left(\mathit{x}\right)$ is called Lyapunov stable if for every ɛ>0 there exists δ>0 so that for any divergence-free vector $\mathit{u}\left(\mathit{x},0\right)\in W^{1+s,p},s>2/p$ , such that ${\Vert \mathit{u}(\mathit{x},0)\Vert }_{L^2}<\delta$ the unique solution $\mathit{u}\left(\mathit{x},t\right)$ to [14]–[16] satisfies

$\begin{array}{ccc}{\Vert \mathit{u}(\mathit{x},t)\Vert }_{L^2}<\varepsilon & \text{for}& t\in [0,\infty )\end{array}$

We note that we require the initial value $\mathit{u}\left(\mathit{x},0\right)$ to be in the Sobolev space $W^{1+s,p},s>p/2$ , since it is known that the 2D Euler equations are globally in time well posed in this function space.

Definition 5

Any steady flow ${\mathit{U}}_0\left(\mathit{x}\right)$ for which the conditions of Definition 4 are violated is called nonlinearly unstable in L ².

Observe that the open issues (in three dimensions) of nonuniqueness or nonexistence of solutions to [14]–[16] would, under Definition 5, be scenarios for instability.

Theorem 6

(Nonlinear instability for 2D Euler flows). Let ${\mathit{U}}_0\left(\mathit{x}\right)\in C^{\infty }\left(T^2\right)$ be satisfy [12]. Let Λ be the maximal Lyapunov exponent to the ODE $\stackrel{.}{x}=U_0(x)$ . Assume that there exists an eigenvalue λ in the L ² spectrum of the linear operator L _E given by [15] with Reλ>Λ. Then in the sense of Definition 5, ${\mathit{U}}_0\left(\mathit{x}\right)$ is Lyapunov unstable with respect to growth in the L ²-norm.

The proof of this result is given in Vishik and Friedlander (2003) and uses a so-called "bootstrap" argument whose origins can be found in references in that article. We remark that the above result gives nonlinear instability with respect to growth of the energy of a perturbation which seems to be a physically reasonable measure of instability.

In order to apply Theorem 6 to a specific 2D flow it is necessary to know that the linear operator L _E has an eigenvalue with Reλ>Λ. As we have discussed, such knowledge is lacking for a generic flow ${\mathit{U}}_0\left(\mathit{x}\right)$ . Once again, we turn to shear flows. As we noted Λ=0 for shear flows, any shear profile for which unstable eigenvalues have been proved to exist provides an example of nonlinear instability with respect to growth in the energy.

We conclude with the observation that it is tempting to speculate that, given the complexity of flows in three dimensions, most, if not all, such inviscid flows are nonlinearly unstable. It is clear from the concept of the fluid Lyapunov exponent that stretching in a flow is associated with instabilities and there are more mechanisms for stretching in three, as opposed to two, dimensions. However, to date there are virtually no mathematical results for the nonlinear stability problem for fully 3D flows and many challenging issues remain entirely open.

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KINETICS OF REVERSIBLE CLUSTERING AND BRANCHING

P.G.J. VAN DONGEN , M.H. ERNST , in Kinetics of Aggregation and Gelation, 1984

2 EQUILIBRIUM SIZE DISTRIBUTION

Following the arguments of [3] we may construct simultaneously the explicit form of the fragmentation rates and the general form of the equilibrium solutions c _km(∞). The ratio of rate constants in the steady state is determined by the Boltzmann factors

(5) $\begin{array}{l}{\text{κ}}_{\text{E}\text{E}}/{\text{λ}}_{\text{E}\text{E}}=\text{exp}\left({\text{ε}}_{\text{E}\text{E}}/\text{k}\text{T}\right)~\text{α}\\ {\text{κ}}_{\text{E}\text{E}}{\text{κ}}_{\text{E}\text{N}}/{\text{λ}}_{\text{E}\text{E}}{\text{λ}}_{\text{E}\text{N}}=\text{exp}\left[\left({\text{ε}}_{\text{E}\text{E}}+{\text{ε}}_{\text{E}\text{N}}\right)/\text{k}\text{T}\right]~\text{β}\end{array}$

which determine in turn the equilibrium solution:

(6) ${\text{c}}_{\text{k}\text{m}}(\text{∞})={\text{N}}_{\text{k}\text{m}}{\text{ζ}}^{\text{k}}{{}_{\text{α}}}^{{\text{n}}_{\text{k}{\text{m}}_{{}_{\text{β}}}\text{m}}}$

The constant ζ is fixed by the constraint $\text{Σ}\text{k}{\text{c}}_{\text{k}\text{m}}=\text{M}=1$ , and the (as yet unknown) combinatorial factors with N₁₀= 1 represent the number of distinct possibilities for constructing a (k,m)-cluster out of k units, given that functional groups are distinguishable and units are not. To further determine the N_km we impose the detailed balance condition, which can be obtained by replacing c_km(t) in (4) by c_km(∞) and setting the individual summands equal to zero, i.e. {… EE …} = o and {… EN …} = o. Next we calculate Σ′ {… EE …} and Σ″ {… EN …}, and eliminate F^EE and F^EN by means of (3), yielding two sets of recursion relations for c_km(∞) or N_km, namely:

(7) $\begin{array}{l}{\text{n}}_{\text{k}\text{m}}{\text{N}}_{\text{k}\text{m}}=\frac{1}{2}\sum \text{'}{\text{e}}_{{\text{k}}_1{\text{m}}_1}{\text{e}}_{{\text{k}}_2{\text{m}}_2}{\text{N}}_{{\text{k}}_1{\text{m}}_1}{\text{N}}_{{\text{k}}_2{\text{m}}_2}\\ 3\text{m}{\text{N}}_{\text{k}\text{m}}=\sum ″{\text{e}}_{{\text{k}}_1{\text{m}}_1}{\text{n}}_{{\text{k}}_2{\text{m}}_2}{\text{N}}_{{\text{k}}_1{\text{m}}_1}{\text{N}}_{{\text{k}}_2{\text{m}}_2}\end{array}$

with N₁₀= 1. The numbers N_km are vanishing whenever n_km= k−2m−1 becomes negative.

Equations (7) have a simple combinatorial interpretation. The first recursion relation states that every (k,m)–configuration is formed n_km= k−2m−l times, once for every node, when nodes are formed between all possible (k₁m₁)– and (k₂m₂)–configurations. Analogously, the second recursion relation states that in the process of formation of all possible branchpoints, every (k,m)–configuration is constructed 3m times, once for every group contained in a branchpoint. The combinatorial factors N_km may be calculated from (8) by using a generating function technique. The result is:

(8) $\begin{array}{l}{\text{N}}_{\text{k}\text{m}}={\left(\text{m}+2\right)}^{-1}{2}^{\text{k}-\text{m}}{\text{Q}}_{\text{k}\text{m}}\\ ={\left[\left(\text{m}+1\right)\left(\text{m}+2\right)\right]}^{-1}{2}^{\text{k}-\text{m}}\left(\begin{array}{c}\text{k}-1\\ 2\text{m}\end{array}\right)\left(\begin{array}{c}2\text{m}\\ \text{m}\end{array}\right)\end{array}$

where Q_km is the counting factor for rooted tree graphs, explicitly calculated in [1]. This model shows a gelation transition of the Flory-Stockmayer type with classical critical exponents, as extensively discussed in [1].

After having determined the combinatorial factors N_km one may express the fragmentation rates F in terms of these N_km with the help of the detailed balance condition.

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Systems of linear differential equations

Henry J. Ricardo , in A Modern Introduction to Differential Equations (Third Edition), 2021

6.11.2 Analysis

Is this the behavior expected from a bouncing mass? First of all, note that the origin is a special point, an equilibrium solution, because both equations of our system vanish at $(x_1,x_2)=(0,0)$ . Physically, this means that a mass-spring system that starts at its equilibrium position $(x(0)=0)$ and has no initial push or pull $(\dot{x}(0)=x_2(0)=0)$ will remain at rest forever, which makes sense.

Now we look closely at a typical closed orbit, as one of these elliptical trajectories is called. Assume that $x_1=0$ and $x_2$ is positive—that is, the mass is at its equilibrium position and is given an initial tug downward. When the mass is at rest $(x_1=0)$ and it is pushed or pulled in a downward direction $(d{x}_1/dt=x_2>0)$ , the flow moves in a clockwise direction (note the direction of the slope field arrows), with $x_2$ decreasing and $x_1$ increasing until the trajectory is at the $x_1$ -axis. Physically, this means that the mass moves downward until the spring reaches its maximum extension ( $x_1$ is at its most positive value), depending on how much force was applied initially to pull the mass downward, at which time the mass has lost all its initial velocity ( $x_2=0$ ). Then the energy stored in the spring serves to pull the mass back up toward its equilibrium position, so that $x_1$ is decreasing at the same time that the velocity $x_2$ is increasing—but in a negative direction (upward). Graphically, this is taking place in the fourth quadrant of the phase plane. When the flow has reached the state $(0,x_2)$ , where $x_2$ is negative, the mass has reached its original position and has attained its maximum velocity upward.

As the trajectory takes us into the third quadrant, the mass is overshooting its original position but is slowing down: $x_1<0$ and $x_2<0$ . When the trajectory has reached the point $(x_1,0)$ , where $x_1$ is negative, the spring is most compressed and the mass is (for an instant) not moving.

As the trajectory moves through the second quadrant, the mass is headed back toward its initial position with increasing velocity in a downward (positive) direction: $x_1<0$ and $x_2>0$ . Finally, the mass reaches its initial position with its initial velocity in the positive (downward) direction— $x_1=0$ , $x_2>0$ —and the cycle begins all over again.

This analysis seems to say that the mass will never stop, bobbing up and down forever. This apparently nonsensical conclusion is perfectly reasonable when you realize that a real mass-spring system is always subject to some air resistance and some sort of friction that slows the system down and eventually forces the mass to stop moving. Our analysis assumes no such impeding force, so the conclusion is rational, even though the assumption is unrealistic.

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Thermodynamics in Chemistry

G.K. Vemulapalli , in Philosophy of Chemistry, 2012

3.7 Chemical Equilibrium in Solution

The expression for the gas phase equilibrium constant is deduced from Gibbs' equation with the aid of the ideal gas equation of state. For solution equilibria, Raoult's and Henry's laws ¹¹ are used to deduce the equilibrium constant expression. As a result concentrations of reactants and products, or their mole fractions appear in the equilibrium constant expression, rather than the partial pressures. However, the solute-solvent interactions are so pronounced that solution equilibrium constants depend on concentrations, which should not be the case if they are true constants. Moreover, the equilibrium constant cannot be estimated from heat capacity and enthalpy data as is done in the gas phase. Considering that application of theory has been seriously compromised and equilibrium constants cannot be estimated prior to measurement, we may wonder about the utility of theory. Again we have to look at the experimental landscape to see why solution thermodynamics continues to thrive.

The aim of the experimental program is now different. First, we note that chemical reactions in solution proceed to equilibrium readily without the aid of a catalyst, Equilibrium studies in the solution phase, compared to those in the gas phase, are relatively easy. Instead of estimating the equilibrium constant from thermo-chemical data, chemists now use equilibrium constant data at different temperatures to estimate the enthalpy and entropy of the reaction. For instance, if we know the equilibrium constant values for the reaction A + B ⇄ AB at two different temperatures, we can estimate enthalpy and entropy of bond formation between A and B. Being aware of the limitations of the approximations used in extending the basic theory this far, chemists accept this sort of data with caution. However, if the research program calls for investigation of a series of related compounds, the data from these studies turn out to be valuable guides in developing models for molecular interactions and complex formation. Here we have another example of how thermodynamics, a theory solely concerned with energy transformation, still provides remarkably useful avenues for research while staying aloof from the approximations needed to study materials.

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Characterisation of Porous Solids V

A. Deryło-Marczewska , ... A. Świątkowski , in Studies in Surface Science and Catalysis, 2000

2.2 Experimental methods

2.2.1 Measurements of benzene adsorption/desorption isotherms

The adsorption/desorption isotherms of benzene vapors were measured at 293   K by gravimetric method using the McBain-Bakr balance.

2.2.2 Adsorption from binary liquid mixtures

The specific surface excess isotherms for binary liquid mixtures: benzene   +   n-heptane and benzene   +   2-propanol were measured by static immersion method [8 ]. The concentrations of equilibrium solutions were determined using HP 5890 gas chromatograph from Hewlett-Packard. The initial mixtures over the whole concentration range served for detector calibration. The surface excess of a given component was calculated from the relation:

(1) $n_i^e=n^o\left(x_i^o-x_i^l\right)/m$

where x _i ^o and x _i ^l , are the mole fractions of component "i" in the initial and equilibrium solutions, respectively, n^o is the initial number of moles in contact with adsorbent of mass m.

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KINETICS OF RED BLOOD CELL AGGREGATION: AN EXAMPLE OF GEOMETRIC POLYMERIZATION*

Alan S. PERELSON , Richard W. SAMSEL , in Kinetics of Aggregation and Gelation, 1984

5.7 Equilibrium Solution

Setting the time derivatives to zero in Eq. (34) allows one to find the equilibrium values of P_f, P_w, and p and hence the equilibrium values of all the macroscopic variables E, M, W, S, T, and R, and the mean quantities given in Eq. (36) . Detailed studies of the equilibrium solution are presented elsewhere ( 4).

When k_w/k_c is large and the reverse rate constant k_r low, our numerical studies show that aggregates grow infinitely large (Fig. 3) signaling the occurrence of a sol-gel phase transition. If we use our model to compute equilibrium solutions past the gel point, we find the concentration of rouleaux is negative. The reason is simple: if the total concentration of caps is positive, the model predicts that cap-cap or cap-wall adhesions occur, thus reducing the number of free rouleaux even if all cells are in one infinite aggregate. From Eqs. (16)–(18)and (20)–(22)one finds

FIGURE 3. Equilibrium solution of the reversible rouleaux equations exhibit a divergence similar to a sol-gel transformation when k′ = k_ww/k_c a_c reaches a critical value. Here ${\text{k}}_{\stackrel{\text{'}}{\text{r}}}={\text{k}}_{\text{r}}/{\text{k}}_{\text{c}}{\text{a}}_{\text{c}}{\text{E}}_0=0.1$ . Note in (a) that both &lt;n&gt;_r, the mean number of cells per rouleau, and, the mean number of branches per rouleau, → ∞ as the nondimensional equilibrium concentration of rouleaux R′ = R/E₀ → 0 in (b). (Reproduced from Ref. (4) with permission).

(37) $\text{d}\text{R}/\text{d}\text{t}=\frac{1}{2}{\text{k}}_{\text{e}\text{e}}{\text{a}}_{\text{e}}{\text{E}}^2-\frac{1}{2}{\text{k}}_{\text{c}\text{c}}{\text{a}}_{\text{c}}{\text{M}}^2-{\text{k}}_{\text{c}\text{w}}\text{M}\text{W}+{\text{k}}_{\text{r}}\left({\text{E}}_0-\text{E}-\text{M}-\text{R}\right).$

Thus as E and R approach zero with M > 0 the negative terms dominate, and dR/dt < 0. We thus can use R < 0 as a criteria for being past the gel point.

In our experiments we generally do not see infinitely large aggregates. Rather the concentration of free caps appears to be reduced by loop formation.

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Given the Ode Find the Constant Solutions and Its Stability

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