PDE notes: heat equation

April 26, 2022June 1, 2022Michael LiuLeave a comment

1. Heat equation

1.1. Introduction

The heat equation corresponding to no sources and constant thermal properties is given as

$\displaystyle u_t = ku_{xx} \ \ \ \ \ (1)$

Equation (1) describes how heat energy spreads out. Other physical quantities besides temperature smooth out in much the same manner, satisfying the same partial differential equation (1). For this reason, (1) is also called the diffusion equation.

Since the heat equation in (1) has one time derivative, we must be given one initial condition (IC) (usually at t = 0), the initial temperature. It is possible that the initial temperature is not constant, but depends on x. Thus, we must be given the initial temperature distribution,

$\displaystyle u(x,0)= f(x) \ \ \ \ \ (2)$

We also need to know that happens at the two boundaries, ${x = 0}$ and ${x = L}$ . Without knowing this information, we cannot predict the future. Two conditions are needed corresponding to the second spatial derivatives present in (1), usually one condition at each end. We call them boundary conditions. We deal with mainly four types of boundary conditions: Dirichlet, Neumann, Robin, and Periodic.

1.2. Separation of Variables (SoV)

When the PDE and the BCs are linear and homogeneous, we use a technique called the method of separation of variables to find the analytic solutions.

We introduce the heat operator,

$\displaystyle L(u)=u_t-ku_{xx} \ \ \ \ \ (3)$

The heat operator is a linear operator that satisfies the linearity property ${L(c_1u_1+c_2u_2)=c_1L(u_1)+c_2L(u_2)}$ .

A linear equation for the unknown ${u}$ is of the form

$\displaystyle L(u)=f \ \ \ \ \ (4)$

where ${L}$ is a linear operator and ${f}$ is known. The heat equation is a linear PDE:

$\displaystyle L(u)=u_t-ku_{xx}=f(x,t) \ \ \ \ \ (5)$

${f(x,t)}$ is a forcing term. If ${f=0}$ , then ${L(u)=0}$ and the PDE becomes

$\displaystyle u_t-ku_{xx}=0 \ \ \ \ \ (6)$

Equation (6) is a linear homogeneous equation.

We now propose to study heat equation with zero temperatures at finite ends:

$\displaystyle \mathrm{PDE: } \hspace{0.5cm} u_t=ku_{xx} \ \ \ \ \ (7)$

$\displaystyle \mathrm{BC1:} \hspace{0.5cm} u(0,t)=0 \ \ \ \ \ (8)$

$\displaystyle \mathrm{BC2:}\hspace{0.5cm} u(L,t)=0 \ \ \ \ \ (9)$

$\displaystyle \mathrm{IC:}\hspace{0.5cm} u(x,0) = f(x) \ \ \ \ \ (10)$

defined for ${0\leq x \leq L}$ and ${t \geq 0 }$ . This problem consists of linear homogeneous partial differential equations with linear homogeneous boundary conditions.

Solution. We attempt to determine the product solution in the form of

$\displaystyle u(x,t)=G(t)\phi(x) \ \ \ \ \ (11)$

First, we substitute equation (11) into equation (7),

$\displaystyle \phi(x)G'(t)=kG(t)\phi^{''}(x) \ \ \ \ \ (12)$

We can “separate variables” by divide both sides of equation (12) by ${kG(t)\phi(x)}$ :

$\displaystyle \frac{1}{kG}G'=\frac{1}{\phi}\phi^{''} \ \ \ \ \ (13)$

Now the variables are “separated” in a sense that the left hand side is only a function of ${t}$ and the right hand side is only a function of ${x}$ . Next, we can claim that both sides of (13) must equal the same constant:

$\displaystyle \frac{1}{kG}G'=\frac{1}{\phi}\phi^{''}=-\lambda \ \ \ \ \ (14)$

where ${\lambda}$ is a constant known as the separation constant.

Equation (14) yields 2 ordinary differential equations,

$\displaystyle G'=-\lambda kG \ \ \ \ \ (15)$

and

$\displaystyle \phi''=-\lambda\phi \ \ \ \ \ (16)$

We now see that the advantage of the product method is that it transform a PDE, which we do not have to solve, into two ODEs. The boundary conditions impose two conditions on the ${x}$ -dependent ODE. The time-dependent ODE, equation (15), has no additional conditions. We can find a general solution for equation (15) quite easily,

$\displaystyle G(t)=ce^{-\lambda kt} \ \ \ \ \ (17)$

where ${c}$ is an arbitrary multiplicative constant. There can be three cases for ${\lambda}$ : ${\lambda>0}$ , ${\lambda=0}$ , ${\lambda<0}$ . We will determine all allowable values of ${\lambda}$ next.

The ODE in (16) has two boundary conditions:

$\displaystyle \phi(0)=0 \ \ \ \ \ (18)$

$\displaystyle \phi(L)=0 \ \ \ \ \ (19)$

We call this a boundary value problem for ordinary differential equations. We note that ${\phi(x)=0}$ satisfy the ODE (16) and both BCs (18) and (19) regardless of the value of the separation constant. Thus, ${\phi(x)=0}$ is a trivial solution of our PDE. It corresponds to ${u(x,t)\equiv0}$ since ${u(x,t)=\phi(x)G(t)}$ . However, we want to see if there are other nontrivial solutions. We will show that there are certain values of ${\lambda}$ , called eigenvalues of the boundary value problem, for which there are nontrivial solutions, ${\phi(x)}$ . The non-trivial solutions are called eigenfunctions corresponding to the eigenvalues ${\lambda}$

Now we solve equation (16). For linear and homogeneous second-order ODE, two independent solutions are usually obtained in the form of ${\phi=e^{rx}}$ . Substituting this result into equation (16) yields

$\displaystyle \phi''=r^2e^{rx} \ \ \ \ \ (20)$

and this implies that

$\displaystyle r^2=-\lambda \ \ \ \ \ (21)$

We consider the three cases of ${\lambda}$ separately.

If ${\lambda<0}$ , the roots of the characteristic polynomials are ${r=\pm\sqrt{-\lambda}}$ , so the solutions are ${e^{\sqrt{-\lambda}x}}$ and ${e^{-\sqrt{-\lambda}x}}$ . The general solution is

$\displaystyle \phi(x)=c_1e^{\sqrt{-\lambda}x}+c_2e^{-\sqrt{-\lambda}x} \ \ \ \ \ (22)$

We use the hyperbolic functions,

$\displaystyle \phi(x)=c_3\mathrm{cosh}(\sqrt{-\lambda}x)+c_4\mathrm{sinh}(\sqrt{-\lambda}x) \ \ \ \ \ (23)$

We consider the BC from (18). Since ${\mathrm{cosh}(0)=1}$ , ${\phi(0)=0}$ implies that ${c_3=0}$ . So we have ${\phi(x)=c_4\mathrm{sinh}(\sqrt{-\lambda}x)}$ . Now we consider ${\phi(L)=0}$ , which implies that ${c_4\mathrm{sinh}(\sqrt{-\lambda}L)=0}$ . Since ${\sqrt{-\lambda}L>0}$ , ${\mathrm{sinh}(\sqrt{-\lambda}L)>0}$ . It follows that ${c_4=0}$ . This implies that ${\phi(x)=0}$ . The only solution of (16) for ${\lambda<0}$ that solves the homogeneous boundary condition is the trivial solution. Thus there are no negative eigenvalues.

We now consider the case of ${\lambda=0}$ . The general solution is

$\displaystyle \phi(x)=c_1+c_2x \ \ \ \ \ (24)$

Apply ${\phi(0)=0}$ and we arrive at ${c_1=0}$ . Apply ${\phi(L)=0}$ , we have ${c_2x=0}$ which implies that ${c_2=0}$ . This implies that ${\phi(x)=0}$ . The only solution of (16) for ${\lambda=0}$ that solves the homogeneous boundary condition is the trivial solution. Thus ${\lambda=0}$ is not an eigenvalue for this problem.

We now consider the case of ${\lambda>0}$ . The roots of the characteristic polynomials are ${r=\pm\sqrt{\lambda}i}$ , so the solutions are ${e^{\sqrt{\lambda}ix}}$ and ${e^{-\sqrt{\lambda}ix}}$ . The general solution is

$\displaystyle \phi(x)=c_1e^{\sqrt{\lambda}ix}+c_2e^{-\sqrt{\lambda}ix} \ \ \ \ \ (25)$

Since ${\mathrm{cos}(\sqrt{\lambda}x)}$ and ${\mathrm{sin}(\sqrt{\lambda}x)}$ are each linear combinations of ${e^{\sqrt{\lambda}ix}}$ and ${e^{-\sqrt{\lambda}ix}}$ , the general solution can also be expresed in the form of

$\displaystyle \phi(x)=c_3\mathrm{cos}(\sqrt{\lambda}x)+c_4\mathrm{sin}(\sqrt{\lambda}x) \ \ \ \ \ (26)$

For the BC ${\phi(0)=0}$ , we have ${\phi(0)=c_3\mathrm{cos}(\sqrt{\lambda}x)}$ which implies that ${c_3=0}$ . Consider the BC ${\phi(L)=0}$ , we have ${c_4\mathrm{sin}(\sqrt{\lambda}x)=0}$ . If we let ${c_4=0}$ , then we arrive at ${u(x,t)=0}$ , a trivial solution. To find the nontrivial solutions, the eigenvalue ${\lambda}$ much satisfy

$\displaystyle \mathrm{sin}(\sqrt{\lambda}L)=0 \ \ \ \ \ (27)$

${\sqrt{\lambda}x}$ must be the zero of the sine function. Thus,

$\displaystyle \sqrt{\lambda}L=n\pi \ \ \ \ \ (28)$

The eigenvalues are

$\displaystyle \lambda=(\frac{n\pi}{L})^2, n=0,1,2,3... \ \ \ \ \ (29)$

The eigenfunctions corresponding to the eigenvalues are

$\displaystyle \phi(x)=c_4\mathrm{sin}(\frac{n\pi x}{L}), n=0,1,2,3... \ \ \ \ \ (30)$

Now we know that the product solution consists of ${\phi(x)=\\c_4\mathrm{sin}(\frac{n\pi x}{L})}$ and ${G(t)=e^{-(\frac{n\pi}{L})^2 kt}}$ where we determined from the boundary conditions ${\phi(0)=0}$ and ${\phi(L)=0}$ . We call this type of boundary conditions the Dirichlet boundary conditions. Thus, the product solutions of the heat equation with homogeneous Dirichlet boundary conditions are

$\displaystyle u(x,t)= B\mathrm{sin}(\frac{n\pi x}{L})e^{-(\frac{n\pi}{L})^2 kt}, n=0,1,2,3... \ \ \ \ \ (31)$

where B is an arbitrary constant.

1.3. Initial value problems

The principle of superposition can be extended to show that if ${u_1 , u_2 , u_3 , . . . , u_M}$ are solutions of a linear homogeneous problem, then any linear combination of these is also a solution, ${\sum_{n=1}^{M} c_nu_n}$ ,where ${c_n}$ are arbitrary constants. Since we know from the method of separation of variables that ${\mathrm{sin}(\frac{n\pi x}{L})e^{-(\frac{n\pi}{L})^2 kt}}$ is a solution of the heat equation (solving zero boundary conditions) for all positive ${n}$ , it follows that any linear combination of these solutions is also a solution of the linear homogeneous heat equation. Thus,

$\displaystyle u(t,x)=\sum_{n=1}^{M} B_n\mathrm{sin}(\frac{n\pi x}{L})e^{-(\frac{n\pi}{L})^2 kt} \ \ \ \ \ (32)$

For example, we wish to solve the following initial value problem:

$\displaystyle \mathrm{PDE: } \hspace{0.5cm} u_t = ku_{xx} \ \ \ \ \ (33)$

$\displaystyle \mathrm{BC1:} \hspace{0.5cm}u(0,t)=0 \ \ \ \ \ (34)$

$\displaystyle \mathrm{BC2:}\hspace{0.5cm}u(L,t)=0 \ \ \ \ \ (35)$

$\displaystyle \mathrm{IC:}\hspace{0.5cm} u(x,0) = 3\mathrm{sin}(\frac{4\pi x}{L})+7\mathrm{sin}(\frac{9\pi x}{L}) \ \ \ \ \ (36)$

Solution. We can let ${u_1(x,0) = 3\mathrm{sin}(\frac{4\pi x}{L})}$ and ${u_2(x,0) = 7\mathrm{sin}(\frac{9\pi x}{L})}$ such that ${u(x,0)=u_1(x,0)+u_2(x,0)}$ . We can use the product solutions from (31) and we see that the product solution satisfy the initial conditions ${B\mathrm{sin}(\frac{n\pi x}{L})}$ . By picking ${B=3}$ and ${n=4}$ , we have satisfied the initial condition ${u_1(x,0)}$ . The solution we get is

$\displaystyle u_1(t,x)=3\mathrm{sin}(\frac{4\pi x}{L})e^{-(\frac{4\pi}{L})^2 kt} \ \ \ \ \ (37)$

By picking ${B=7}$ and ${n=9}$ , we have satisfied the initial condition ${u_2(x,0)}$ . The solution we get is

$\displaystyle u_1(t,x)=7\mathrm{sin}(\frac{9\pi x}{L})e^{-(\frac{9\pi}{L})^2 kt} \ \ \ \ \ (38)$

By superposition principle we know that the product solution of our problem is a linear combinations of ${u_1(x,t)}$ and ${u_2(x,t)}$ . Thus,

$\displaystyle u(t,x)=3\mathrm{sin}(\frac{4\pi x}{L})e^{-(\frac{4\pi}{L})^2 kt}+ 7\mathrm{sin}(\frac{9\pi x}{L})e^{-(\frac{9\pi}{L})^2 kt} \ \ \ \ \ (39)$

${\square}$

The IC in the previous problem is a finite sum of the sine functions. What should we do in the situation that the IC is not a finite linear combinations of the sine functions? Consider the following problem:

$\displaystyle \mathrm{PDE: } \hspace{0.5cm} u_t = ku_{xx} \ \ \ \ \ (40)$

$\displaystyle \mathrm{BC1:} \hspace{0.5cm}u(0,t)=0 \ \ \ \ \ (41)$

$\displaystyle \mathrm{BC2:}\hspace{0.5cm}u(L,t)=0 \ \ \ \ \ (42)$

$\displaystyle \mathrm{IC:}\hspace{0.5cm} u(x,0) = 100 \ \ \ \ \ (43)$

Solution. From equation (32) we know that the solution is

$\displaystyle u(t,x)=\sum_{n=1}^{M} B_n\mathrm{sin}(\frac{n\pi x}{L})e^{-(\frac{n\pi}{L})^2 kt} \ \ \ \ \ (44)$

We want to apply the IC to our problem. First, from the previous problem, note that we can solve the heat equation if initially

$\displaystyle u(x,0)= f(x) = \sum_{n=1}^{M} B_n\mathrm{sin}(\frac{n\pi x}{L}) \ \ \ \ \ (45)$

Thus, we use the theory of Fourier series and claim that “any” (with restrictions) initial condition ${f(x)}$ can be written as an infinite linear combination of ${\mathrm{sin}(\frac{n\pi x}{L})}$ , known as a type of Fourier series:

$\displaystyle f(x) = \sum_{n=1}^{\infty} B_n\mathrm{sin}(\frac{n\pi x}{L}) \ \ \ \ \ (46)$

We want to determine the coefficient ${B_n}$ in equation (46). To do that, we use the fact the ${\mathrm{sin}(\frac{n\pi x}{L})}$ satisfies the following orthogonality relation:

$\displaystyle \int_{0}^{L} \mathrm{sin}(\frac{n \pi x}{L})\mathrm{sin}(\frac{m \pi x}{L}) \,dx = \left\{ \begin{array}{lr} 0 & n \neq m \\ \frac{L}{2} & n=m \\ \end{array} \right. \ \ \ \ \ (47)$

Multiply (46) by ${\mathrm{sin}(\frac{m \pi x}{L})}$ and integrating from ${0}$ to ${L}$ yields

$\displaystyle \int_{0}^{L} f(x)\mathrm{sin}(\frac{m \pi x}{L})\, dx = B_{m}\int_{0}^{L}\mathrm{sin^2}(\frac{m\pi x}{L})\, dx \ \ \ \ \ (48)$

Solving for ${A_m}$ yields,

$\displaystyle B_m=\frac{2}{L} \int_{0}^{L}f(x)\mathrm{sin}(\frac{m \pi x}{L})\, dx \ \ \ \ \ (49)$

Now, we calculate the coefficient ${B_n}$ from equation (49) for ${f(x)=100}$ ,

$\displaystyle B_n=\frac{2}{L} \int_{0}^{L}100\mathrm{sin}(\frac{n \pi x}{L})\, dx \ \ \ \ \ (50)$

Solving for ${B_n}$ yields,

$\displaystyle \left\{ \begin{array}{lr} 0 & n \hspace{5pt} even\\ \frac{400}{n\pi} & n\hspace{5pt} odd \\ \end{array} \right. \ \ \ \ \ (51)$

Each succeeding term in the series is much smaller than the first. We can then approximate the infinite series by only the first term:

$\displaystyle u(t,x)\approx \frac{400}{\pi}\mathrm{sin}(\frac{\pi x}{L})e^{-(\frac{\pi}{L})^2 kt} \ \ \ \ \ (52)$

${\square}$

Let us summarize the method of separation of variables as it appears for the heat equation with homogeneous Dirichlet boundary conditions:

$\displaystyle \mathrm{PDE: } \hspace{0.5cm} u_t = ku_{xx} \ \ \ \ \ (53)$

$\displaystyle \mathrm{BC1:} \hspace{0.5cm}u(0,t)=0 \ \ \ \ \ (54)$

$\displaystyle \mathrm{BC2:}\hspace{0.5cm}u(L,t)=0 \ \ \ \ \ (55)$

$\displaystyle \mathrm{IC:}\hspace{0.5cm} u(x,0) = f(x) \ \ \ \ \ (56)$

Make sure that you have a linear and homogeneous PDE with linear and homogeneous BC.
Temporarily ignore the nonzero IC.
Separate variables (determine differential equations implied by the assumption of product solutions) and introduce a separation constant.
Determine separation constants as the eigenvalues of a boundary value problem.
Solve other differential equations. Record all product solutions of the PDE obtainable by this method.
Apply the principle of superposition (for a linear combination of all product solutions).
Attempt to satisfy the initial condition.
Determine coefficients using the orthogonality of the eigenfunctions.

1.4. Other Boundary Value Problems

The following problem is defined for ${0\leq x \leq L}$ and ${t \geq 0 }$ .

$\displaystyle \mathrm{PDE: } \hspace{0.5cm} u_t = ku_{xx} \ \ \ \ \ (57)$

$\displaystyle \mathrm{BC1:} \hspace{0.5cm}u_x(0,t)=0 \ \ \ \ \ (58)$

$\displaystyle \mathrm{BC2:}\hspace{0.5cm}u_x(L,t)=0 \ \ \ \ \ (59)$

$\displaystyle \mathrm{IC:}\hspace{0.5cm} u(x,0) = f(x) \ \ \ \ \ (60)$

This is a heat conduction problem in a one-dimensional rod with constant thermal properties and no sources. The ends of the rod are insulated. Both the PDE and BCs are linear and homogeneous. We call this type of BC the Neumann boundary condition. We apply SoV.

Solution. The assumed product solutions are:

$\displaystyle u(x,t)=\phi(x)G(t) \ \ \ \ \ (61)$

Substitute into the PDE,

$\displaystyle \phi G'= kG\phi^{''}= -\lambda \ \ \ \ \ (62)$

where ${\lambda}$ is the separation constant. This implies that

$\displaystyle G^{'}=-\lambda kG \ \ \ \ \ (63)$

$\displaystyle \phi^{''}=-\lambda\phi \ \ \ \ \ (64)$

Solving (64),

$\displaystyle G(t)=e^{-\lambda kt} \ \ \ \ \ (65)$

The insulated BCs imply that the separated solutions must satisfy

$\displaystyle \phi^{'}(0)=0 \ \ \ \ \ (66)$

$\displaystyle \phi^{'}(L)=0 \ \ \ \ \ (67)$

We determine the separation constant ${\lambda}$ by finding those ${\lambda}$ for which nontrivial solutions exist.

For ${\lambda <0}$ , the general solution is

$\displaystyle \phi(x)=c_1e^{\sqrt{-\lambda}x}+c_2e^{-\sqrt{-\lambda}x} \ \ \ \ \ (68)$

We also need

$\displaystyle \phi^{'}(x)=\sqrt{-\lambda}(c_1e^{\sqrt{-\lambda}x}-c_2e^{-\sqrt{-\lambda}x}) \ \ \ \ \ (69)$

The BC ${\phi^{'}(0)=0}$ implies that ${c_1=c_2}$ , and ${\phi^{'}(L)=0}$ means that

$\displaystyle \sqrt{-\lambda}(c_1e^{\sqrt{-\lambda}L}-c_2e^{-\sqrt{-\lambda}L})=2\sqrt{-\lambda}c_1\mathrm{sinh}(\sqrt{-\lambda} L)=0 \ \ \ \ \ (70)$

Since ${\sqrt{-\lambda} L >0}$ , ${\mathrm{sinh}(\sqrt{-\lambda} L)>0}$ . This implies that ${c_1=0}$ . Since ${\phi(x)=0}$ is a trivial solution, there is no negative eigenvalues.

For ${\lambda=0}$ , the solution is

$\displaystyle \phi(x)=c_1+c_2x \ \ \ \ \ (71)$

and

$\displaystyle \phi^{'}(x)=c_2 \ \ \ \ \ (72)$

Both BCs give ${c_2=0}$ . Thus, there are non trivial solutions of the BVP for ${\lambda=0}$ , namely,

$\displaystyle \phi(x)=c_1 \ \ \ \ \ (73)$

The time dependent part gives ${G(t)=e^{-\lambda kt}=1}$ . Thus, the resulting product solution of the PDE is

$\displaystyle u(x,t)=A \ \ \ \ \ (74)$

where ${A}$ is any constant.

For ${\lambda>0}$ , the solutions is in the form

$\displaystyle \phi(x)=e^{\pm\sqrt{\lambda}ix} \ \ \ \ \ (75)$

which gives

$\displaystyle \phi(x)=c_1\mathrm{cos}(\sqrt{\lambda} x)+c_2\mathrm{sin}(\sqrt{\lambda} x) \ \ \ \ \ (76)$

and

$\displaystyle \phi^{'}(x)=\sqrt{\lambda}(c_2\mathrm{cos}(\sqrt{\lambda} x)-c_1\mathrm{sin}(\sqrt{\lambda} x)) \ \ \ \ \ (77)$

The BC ${\phi^{'}(0)=0}$ implies that ${c_2\sqrt{\lambda}=0}$ , so ${c_2=0}$ . The BC ${\phi^{'}(L)=0}$ gives

$\displaystyle -\sqrt{\lambda}c_1\mathrm{sin}(\sqrt{\lambda}L)=0 \ \ \ \ \ (78)$

For nontrivial solutions, ${c_1\neq0}$ . This gives us ${\mathrm{sin}(\sqrt{\lambda}L)=0}$ .

$\displaystyle \lambda= (\frac{n\pi}{L})^2, n=0,1,2,3... \ \ \ \ \ (79)$

However, the corresponding eigenfunctions are cosines,

$\displaystyle \phi(x)=c_1\mathrm{cos}(\frac{n\pi x}{L}), n=0,1,2,3... \ \ \ \ \ (80)$

The product solution of the PDE is

$\displaystyle u(t,x)=Ae^{-(\frac{n\pi}{L})^2kt}\mathrm{cos}(\frac{n\pi x}{L}), n=0,1,2,3... \ \ \ \ \ (81)$

where A is a arbitrary multiplicative constant.

In order to satisfy the IC, we use the principle of superposition so that,

$\displaystyle u(t,x)=A_0+\sum_{n=1}^{\infty}A_{n}e^{-(\frac{n\pi}{L})^2kt}\mathrm{cos}(\frac{n\pi x}{L}) \ \ \ \ \ (82)$

The IE ${u(x,0)=f(x)}$ is satisfied if

$\displaystyle f(x)=A_0+\sum_{n=1}^{\infty}A_{n}\mathrm{cos}(\frac{n\pi x}{L}) \ \ \ \ \ (83)$

for ${0\leq x \leq L}$ .

To complete the solution, we need to determine the arbitrary coefficient ${A_0}$ and ${A_n}$ . We use the fact that ${\mathrm{cos}(\frac{n\pi x}{L})}$ satisfies the following orthogonality relation:

$\displaystyle \int_{0}^{L} \mathrm{cos}(\frac{n \pi x}{L})\mathrm{cos}(\frac{m \pi x}{L}) \,dx = \left\{ \begin{array}{lr} 0 & n \neq m \\ \frac{L}{2} & n=m\neq 0 \\ L & n=m=0 \end{array} \right. \ \ \ \ \ (84)$

for ${n}$ and ${m}$ nonnegative integers.

Multiply (83) by ${\mathrm{cos}(\frac{m \pi x}{L})}$ and integrating from ${0}$ to ${L}$ yields

$\displaystyle \int_{0}^{L} f(x)\mathrm{cos}(\frac{m \pi x}{L})\, dx = A_{m}\int_{0}^{L}\mathrm{cos^2}(\frac{m\pi x}{L})\, dx \ \ \ \ \ (85)$

Solving for ${A_m}$ yields,

$\displaystyle A_0= \frac{1}{L} \int_{0}^{L}f(x)\, dx \ \ \ \ \ (86)$

$\displaystyle A_m=\frac{2}{L} \int_{0}^{L}f(x)\mathrm{cos}(\frac{m \pi x}{L})\, dx , m\geq 1 \ \ \ \ \ (87)$

${\square}$

Percolation theory: percolation on a square lattice

April 11, 2022April 12, 2022Michael LiuLeave a comment

Figure 1. Percolation on a 100×100 square lattice at p=0.3, 0.5, 0,7 (in the order of top to bottom). The red squares are ones that can be reached from the top or bottom by going only on filled squares.

In the previous post, we introduced the basic concepts of percolation theory. Here we define the terminologies formally. Let $E$ denote the edge set of a graph $\wedge$ . We take $\Omega = \prod_{e\in E} \{0,1\}$ as a sample space, points in which are represented as $\omega=(\omega(e):e\in \mathbb{E}^d)$ and are called configurations. A bond $e$ is open in the configuration $\omega$ if $\omega(e)=1$ and is closed if $\omega(e)=0$ , so configurations correspond to open subgraphs. We take $F$ to be the $\sigma-$ field of subsets of $\Omega$ generated by the cylinder sets. The associated probability measure is going to be the product measure with density $p$ on $(\Omega, F)$ ,

$P_p=\prod_{e\in E} \mu_e$

where $\mu_e$ is Bernoulli measure on {0,1}, given by $\mu_p(1)=p$ and $\mu_p(0)=1-p$ . We can take $\Omega$ to be the set of possible outcomes of our random subgraph and take $P_p$ to be describing its distribution. We can basically ignore the measure-theoretic details above but take note that we do use the notation $P_p$ to describe probabilities when the parameter used is $p$ .

Let us recall the definition critical probability, $p_H =$ sup $\{p:\theta(p)=0\} =$ inf $\{p:\theta(p)>0\}$ . We now present a basic result in probability theory.

Theorem 2.1 Suppose that ${X_n}$ is a sequence of independent random variables. Let A be an event in the $\sigma$ -field generated by ${X_n}$ . Suppose that the event A is independent of each finite subset of ${X_n}$ , then $\mathbb{P}(A)$ is 0 or 1 .

This is known as Kolmogorov’s 0-1 law. In percolation, we know that each site or bond has an independent probability $p$ . Let $E$ be an event that there is an infinite open cluster. The event $E$ is invariant under finite changes of sites or bonds. Thus, Kolmogorov’s 0-1 law implies that $\mathbb{P}$ (E) is either 0 or 1.

If $p<p_H$ , then

$P_p(E)\leq \sum_{x}P_p(|C_x|=\infty)=\sum_{x}\theta_x(p)=0$

If $p>p_H$ , then

$P_p(E)\geq P_p(|C_x|=\infty)=\theta_x(p) > 0$

for some site $x$ , implying that $P_p(E)=1$ . In this case, we say that percolation occurs.

Let’s sum up what we have shown so far with an analogy. Consider the case of water moving through a very large medium (so large that it will take infinitely long for water to reach the surface from an origin inside the medium). Let the probability that a pore is big enough for water to pass through be $p$ . If the medium is made of less porous material, the pores having small $p$ are less likely to be open for the water to flow through, so any open path that water travels through will likely be short. However, if it is made of more porous material with higher $p$ , there are more open channels, so water can travel through a longer path from the origin. It is obvious to see that as $p$ increases, it is likely that, eventually, water will travel infinitely far. We have observed that there is a critical probability $p_H$ representing a threshold for such an event to occur. If $p>p_H$ , then there is a positive probability that the path that water travels through is infinite. Kolmogorov’s 0-1 law guarantees that if $p>p_H$ , an infinite path must exist inside the medium, though in our case, water may not flow in that path (as there can be many disconnected open paths inside the medium and, base on our assumption, water only flow in one of them).

Figure 2. A sketch of the structure of a two-dimensional porous medium. The lines indicate open paths. Comparing both graphs, we see that water flows in a shorter path in a less porous medium with a lower value of p.

We see that on either side of the critical probability, the global behavior of the system is fundamentally different. And at critical probability, a sharp transition takes place which transforms the behavior of the system from one form to the other. Thus, the existence of a critical probability makes percolation a mathematically interesting and rich subject.

Next, we continue with our study of percolation on $\mathbb{Z}^2$ . First, we want to show that bond percolation on $\mathbb{Z}^2$ is non-trivial: $0<p_H<1$ . Let $\mu(\wedge;x)$ be the number of self-avoiding walks in $\wedge$ starting at $x$ . In graph theory, a self-avoiding walk is also known as a path. As $\mathbb{Z}^2$ is 4 regular, the number of self-avoiding walks of length $n$ starting at the origin 0 is $\mu_n=\mu_n(\mathbb{Z}^2)=\mu_n(\mathbb{Z}^2;0)\leq 4\times 3^{n-1}$ , since there are 4 choices for the first step and at most 3 choices in the later steps.

Theorem 2.2 For bond percolation in $\mathbb{Z}^2$ , we have $p_H \geq \frac{1}{3}$ .

Proof Let $F_n$ be the event that there is an open cluster $C_0$ of size $n$ starting at the origin 0 where each bond is open with probability $p$ . Then the probability that all bonds are open is $p^n$ . For every site $x \in C_0$ , there is at least one open path from from 0 to $x$ . Thus, we have

$P_p(F_n)\leq \mu_n p^n= \frac{4}{3}(3p^n)$

For $p<\frac{1}{3}, (3p^n)<1$ . Since ${|C_0|=\infty}\subseteq F_n$ $\forall$ $n$, let there be an infinite cluster from the origin with $p<\frac{1}{3}$ , we have

$P_p(|C_0|=\infty)=\lim_{n\rightarrow\infty}P_\textit{p}(F_n)\leq \lim_{n\rightarrow\infty} \frac{4}{3}(3p^n)=0$

This is the equivalent to saying if $p<\frac{1}{3}$ , $\theta(p)=0$ . By definition of $p_H$ , it follows that $p_H \geq \frac{1}{3}$ .

$\square$

We now consider an upper bound. We will make use of a key result from graph theory.

Lemma 2.3 (Grimmett) Let $C$ be a vertex set of a connected subgraph of $\mathbb{Z}^2$ . $|C|<\infty$ if and only if $\exists$ a simple cycle with $C$ in its interior.

No proof will be provided here, but by drawing some pictures we can convince ourselves that it is very believable. Looking at figures 3 and 4 will also be helpful.

Theorem 2.4 For bond percolation in $\mathbb{Z}^2$ , we have $p_H \leq \frac{2}{3}$ .

Proof The first thing we do is to introduce the dual $\wedge^*$ of a graph $\wedge$ drawn in the plane has a vertex for each face of $\wedge$ and an edge $e^*$ for each edge $e$ in $\wedge$ . When $\wedge=\mathbb{Z}^2$ , we take $\wedge^* = \mathbb{Z}^2+(\frac{1}{2},\frac{1}{2})$ , which is isomorphic to $\wedge$ . We call an bond in the dual graph $(\mathbb{Z}^2)^*$ open if and only if the corresponding bond in the original graph $\mathbb{Z}^2$ is closed. If the set of open bonds in $\mathbb{Z}^2$ are given by $P_p$ , then the distribution of the set of closed bonds in $(\mathbb{Z}^2)^*$ will also be given by $P_p$ . An \textit{open dual cycle} is a cycle in the dual graph consisting of dual bonds that are open.

Now that we have defined all the necessary terms. Suppose that $p>\frac{2}{3}$ , let $L_k$ be the line segment joining the origin to the point $(k,0)$ , and let $S$ be a dual cycle surrounding $L_k$ , which has a length $n$ . Then $S$ must contain a dual bond $e^*$ crossing the positive $x$ -axis at some coordinate between $(k+\frac{1}{2},0)$ and $(\frac{1}{2}(n-3),0)$ . This means that we have less than $\frac{n}{2}$ choices for $e^*$ . As the rest of S is a path of length $n-1$ in the dual lattice, we have shown that the number of cycles around the origin of length $n$ is at most $\frac{n}{2}\times 4(3^{n-1})$ . Let $A_n$ be the event that there exists an open dual cycle surrounding $L_k$ , since dual bonds are open with probability $1-p$ , we have

$\sum_{n=4}^{\infty} P_p(A_n) \leq \sum_{n=4}^{\infty}\frac{2n}{3}(3(1-p))^n$

Since $3(1-n)<1$ , the sum is convergent. We can choose a $N$ so that $\sum_{n\geq N}^{\infty}\frac{2n}{3}(3(1-p))^n <1$ . Now, let $E_1$ be the event that the $N$ bonds in the line segment $L_N$ are open. Let $E_2$ be the event that there are no open dual cycles surrounding $L_N$ . Since $\sum_{n=4}^{\infty} P_p(A_n)<1$ , $P_p(E_2)>0$ . If both $E_1$ and $E_2$ holds then $|C|=\infty$ by Lemma 2.3. Since $E_1$ and $E_2$ are independent, we have

$\theta(p)\geq P_p(E_1 \cap E_2) = P_p(E_1)P_p(E_2) = P_p(E_2)p^N >0$

Since $p>\frac{2}{3}$ , by definition of $p_H$ , it follows that $p_H \leq \frac{2}{3}$ .

$\square$

Figure 4. A finite open cluster at the origin, surrounded by an open dual cycle (corresponding to closed bonds in the original lattice)

We have shown that the critical probability $p_H$ for $\mathbb{Z}^2$ is in between $\frac{1}{3}$ and $\frac{2}{3}$ , so there exists a non-trivial critical phenomenon. On the basis of Monte Carlo simulations, it was suggested that the critical probability should be $\frac{1}{2}$ . In the next post, we will show that the critical probability for $\mathbb{Z}^2$ is in fact $\frac{1}{2}$ .

Percolation theory: basic concepts

March 13, 2022April 11, 2022Michael LiuLeave a comment

Note: In this post, I will use standard terminology of graph theory, such as vertex, edge, component, path, subgraph, etc.

Consider the following scenarios:

Suppose a large porous rock is submerged under water for a long time, will the water reach the center of the stone?
How far from each other should trees in an orchard be planted in order to minimize the spread of fire?
How infectious does a strain of flu have to be to create a pandemic? What is the expected size of an outbreak?

The problems above are all related to percolation theory.

The aim of this post is to introduce some basic concepts of percolation theory. Percolation is a simple probabilistic model which exhibits critical phenomena. This means that there is a natural parameter in the model at which the behavior of the system changes drastically. Interpreted narrowly, percolation theory can be taken as the study of the component structure of random subgraphs of graphs.

In the standard model of percolation theory, we consider the underlying graph to be a lattice or lattice-like graph. For example, suppose the graph $\wedge$ consists of the set $\mathbb{Z}^d$ as the vertex set together with an edge between any two points having Euclidean distance 1. For Percolation, we use the term sites instead of vertices and bonds instead of edges. To obtain a random subgraph of $\wedge$ , we let $p \in [0,1]$ be the crucial parameter of our model. Then we select bonds and sites independently with the same probability $p$ . The sited and bonds selected are called open and those not selected are called closed. (In other words, each sites or bonds open with probability $p$ and closed with probability $1-p$ ). When our random subgraph is obtained by keeping the open sites, we speak of site percolation; when we keep the open bonds, bond percolation. In site percolation, the open subgraph is formed by the open sites; in bond percolation, the open subgraph is formed by the open edges and all vertices.

Figure 1. Parts of the open subgraphs in bond percolation on the square lattice $\mathbb{Z}^2$ .

Figure 1. Parts of the open subgraphs in bond percolation on the square lattice $\mathbb{Z}^2$ .

Figure 2. Parts of the open subgraphs in site percolation on the square lattice $\mathbb{Z}^2$ . The filled circles are the open sites.

In general, we assume that $\wedge$ is connected, infinite, and locally finite (i.e., every vertex has finite degree). An open path is a path (i.e., self-avoiding walk) in the open subgraph. For sites $x$ and $y$ in the open subgraph, we write ${x \rightarrow y}$ if there is an open path from $x$ to $y$ . We also write { ${x \rightarrow \infty}$ } if there is an infinite open path starting at $x$ .

Let $C_x$ denote the component (i.e., the set vertices connected to $x$ via an open path) containing $x$ in our random subgraph. In percolation, we call the components of the random subgraph open clusters. Since the graphs we consider are locally finite, an open cluster is infinite if and only if, for every site $x$ in the cluster, { $x \rightarrow \infty$ } holds. If $x$ is not open, then in site percolation, $C_x = \emptyset$ , and in bond percolation, $C_x = {x}$ .

Using figures 1 and 2 as an example: there are 10 open clusters for bond percolation in figure 1, and there are 4 open clusters for site percolation in figure 2.

Now, our first basic question is the following: what is the probability that there exists an open path from an open site $x$ to infinity? This is the same as asking for the probability that $C_x$ is infinite. We denote this probability as $\theta_{x}(p)$ and call it as percolation probability.

More formally, we can use bond percolation as an example. $\theta_{x}(p)=P(x \rightarrow \infty ) = P(|C_x| = \infty )$ , where $|C_x|=|V(C_x)|$ is the number of sites in $C_x$ . Two sites $x$ and $y$ are equivalent if there is an automorphism of $\wedge$ mapping $x$ to $y$ . If we assume that all sites of $\wedge$ are equivalent, then $\theta_{x}(p)$ is the same for all sites $x$ and we can write $\theta_{x}(p)=\theta(p)$ . Clearly, $\theta_{x}(0)=0$ and $\theta_{x}(1)=1$ , since there are no open bonds at all when $p=0$ and all bonds are open when $p=1$ . If $x$ and $y$ are sites at distance $d$ , then $\theta_{x}(p)\geq p^d\theta_{y}(p)$ , so either $\theta_{x}(p)=0$ for every site $x$ , or $\theta_{x}(p)>0$ for every site $x$ . It is also intuitively clear that $\theta(p)$ is an increasing function of $p$ . Thus, the graph of $\theta(p)$ should have the form indicated in figure 3. It now follows that we can define a critical probability $p_H$ (indicated as $p_c$ in figure 3), $0\geq p_H \geq 1$ , by $p_H =$ sup $\{p:\theta(p)=0\}=$ inf ${p:\theta(p)>0}$ . This means that if $p<p_H$ , then $\theta_{x}(p)=0$ $\forall x$ , and if $p>p_H$ , then $\theta_{x}(p)>0$ $\forall x$ . For a graph $\wedge$ , we write $p^{s}_{H}$ for site percolation and $p^{b}_{H}$ for bond percolation.

Figure 3. Graph of $\theta(p)$ with respect to $p$ .

In the next post, we will talk more about critical probability and percolation on $\mathbb{Z}^2$ .

[Upcoming] Exegetical Writing 4: Daodejing – chapter 40; Argumentative Writing 1: Is Mencius right to claim that emotions are the origin of morality? Why or why not? What is the role of reason?

October 1, 2021May 19, 2022Michael LiuLeave a comment

Argumentative Writing 2: Emptiness, Dependent Arising, and the Cosmology of Tiantai School

October 1, 2021January 21, 2022Michael LiuLeave a comment

Emptiness (Śūnyatā, 空) and Dependent Arising (Pratītyasamutpāda, 緣起) are common tenets in all Buddhist schools, but different schools give the terms different interpretations. In this essay, I will examine the philosophical development of the concepts of Emptiness and Dependent Arising in the Tiantai school, and discuss how they have influenced the cosmology of Mahayana Buddhism in East Asia.

The Tiantai school of Buddhism was founded in six century China. The major theses of Tiantai school are established by the fourth patriarch Zhiyi, who regards the Lotus Sutra as having the highest status of all Buddhist doctrines. Zhiyi takes Emptiness and Dependent Arising to be the core concepts of his defining argument – the Threefold Truth (三諦), which includes the truth of Emptiness, the truth of Temporariness, and the truth of the Middle Way. The truth of Emptiness states that all dharmas are empty in the sense that they do not have inherent self-nature. The truth of Temporariness states that all dharmas depend on many causal factors to exist. Without these factors, things would not have come into being. Therefore, dharmas exist, but they exist provisionally, temporally, and dependently. I argue that the first two truths proposed by Zhiyi are similar to Nagarjuna’s theory of Twofold Truths: on the level of the phenomenal world, all things exist independently and provisionally; on the level of ultimate reality (Nirvana), however, things are all empty in nature. (Mitchell and Jacoby, p. 160) Nevertheless, Zhiyi makes an explicit integration of the truth of Emptiness and the truth of Temporariness by constructing a third truth: the truth of the Middle Way. In Fahua xuanyi (法華玄義), Zhiyi explains the notion of Middle as “The reality of non-duality is called the Middle.” The “Way” is translated from the Chinese word “Dao (道)”, which means “the ultimate reality of the universe”. Thus, the truth of the Middle Way simply states that the ultimate reality of the world is that all dharmas are both provisionally existent and empty. Finally, Zhiyi completes his argument by saying, in Fahua xuanyi, “Though these are an integrated unity, they are called threefold; though they are threefold, they are called a unity.” Therefore, the Threefold Truths can be seen as three aspects of one thing – the real nature of our world. In other words, Emptiness, Dependent arising, and the Middle Way are ultimately identical. Standing alone, each of them represents a partial truth; taken together, they jointly depict the one reality.

Zhiyi’s notion of Threefold Truths reveals a fundamental distinction between the cosmology of early Buddhist traditions and Tiantai Buddhism (as well as other major Mahayana schools). Early Buddhism sees the phenomenal world and nirvana as two separate realms of existence. The ordinary sentient beings who live in the phenomenal world go through the endless cycles of life in the six dharma realms. When one is enlightened, he/she will exit the six dharma realms and be completely free from the cycles of life and death. The phenomenal world is not real, since all things are devoid of inherent self-nature and independent existence. The realm of nirvana is the ultimate reality – it is real, unconditioned, and empty. (Mitchell and Jacoby, p. 62)

Figure 1. A painting showing the Cyclic Existence, or Saṃsāra, in early Buddhism

On the other hand, the Tiantai school rejects the duality of this worldview by claiming that there is no other realm outside the provisionally existent phenomenal world as shown in the Threefold Truths. In other words, the phenomenal world is nirvana and nirvana is the phenomenal world, as Zhiyi puts it in Mohe Zhiguan (摩訶止観), “A single, unalloyed reality is all there is.” Zhiyi describes the worldview of Tiantai as having ten dharmas realms. Besides the previous six mundane realms, there is also four holy realms: the realm of Śrāvaka (Hearers of Buddha, 緣覺乘), the realm of Pratyekabuddhayāna (Solitary Buddha, 辟支佛), the realm of bodhisattva (菩薩), and the realm of Buddha. One notable characteristic of this worldview is that the ten dharma realms are mutually contained, meaning that they are all part of the whole world. Thus, all ten dharma realms are at the same time empty, provisionally and dependently existing, and manifesting the Middle Way.

Figure 2. A painting showing the Ten Dharma Realms

Base on this worldview, Zhiyi puts forward another important thesis for the Tantai school:

“One mind contains ten dharma-realms, but each of these dharma-realms contains ten dharma-realms, resulting in 100 dharma-realms. One realm contains 30 kinds of worlds, hence 100 dharma-realms contain 3,000 worlds. These 3,000 worlds are contained in a fleeting moment of thought.” (Mitchell and Jacoby, p.235)

The assumption of this argument is that all of us have Buddha nature – the potential to become a Buddha. Indeed, the Lotus Sutra says that “the Buddhas of the present and future, whose numbers are beyond calculations”. Base on the Tiantai worldview, those Buddhas reside in the same reality as we do. The moment they gain true enlightenment, they turn the phenomenal world into Nirvana. Thus, entering Nirvana depends on nothing more than our own mind’s ability to perceive and understand the reality. The minds of sentient beings conceive all dharmas to be full of defilements and attachments; they are thus stuck in the cycles of life and death and can never be totally free from Duḥkha (suffering). The minds of Śrāvaka, Pratyekabuddhayāna, and bodhisattva are able to conceive all dharmas to be dependently arisen, thus they separate the phenomenal world and Nirvana into two distinct realms. Finally, the minds of Buddha conceive all three thousand worlds to realize their true nature – all things in the world are provisionally and dependently existent, empty in nature, and manifest the Middle Way. Thus, he/she is able to understand Cosmo as one unified reality. Therefore, one can conclude that the creation of three thousand worlds in one thought depends on one’s own mental power – one thought can bring one down to the realm of hell; the other can raise one up to the realm of Buddha.

All in all, the Tiantai school developed a new cosmology for Mahayana Buddhism in East Asia based on the concepts of Emptiness, Dependent Arising, and the Middle Way. By eliminating the barrier between the phenomenal world and the realm of Nirvana, Tiantai Buddhism offers all sentient beings a chance to be free from suffering and become enlightened. Personally, I appreciate the Tiantai tradition for making the very first attempt to transform early Indian Buddhism into a more accessible and inclusive religion. I also found the major arguments made by the Tiantai scholars logical and rigorous, so I am convinced (and impressed) by this development.

Works Cited

Donald W. Mitchell, S. H. (2013). Buddhism: Introducing the Buddhist Experience. Oxford: Oxford University Press.

Argumentative Writing 1: “Gradual enlightenment” vs “Sudden enlightenment” in Chan(Zen) Buddhism

September 28, 2021January 21, 2022Michael LiuLeave a comment

There is a major distinction between the Northern School of Chan and the Southern School of Chan – the Northern School taught gradual enlightenment; The Southern School taught sudden enlightenment. In this essay, I will begin by analyzing the two poems written by Shenxiu and Huineng, to see how the two schools diverge theoretically.

Shenxiu’s poem says (Mou-Lam and Price, p. 72):

Our body is the bodhi tree,

And our mind a mirror bright.

Carefully we wipe them hour by hour,

And let no dust alight.

Shenxiu’s verse starts with an assumption that we have an originally pure mind, which, I argue, is the theoretical foundation of the Northern School. Shenxiu further explores the properties of the mind in his work Guan Xin Lun, “Of the myriad dharmas, the mind is the most basic…There are two different minds…The first is the pure mind. The second is the defiled mind.” The two states of mind can be explained in the following ways: Our mind is initially pure and clean, but it is venerable to the distortions by defilements (dust) caused by “Three Poisons,” namely greed, anger, and ignorance. The purpose of Buddhist practice is thus to purify our mind and get rid of all defilements. In particular, Shenxiu’s teaching focuses on “beholding the mind (guanxin),” which means to keep a vigilant watch over the mind through solitary meditation. By keeping our mind and our initial purity in view, one can identify and avoid all sorts of defilements, and thus “let no dust alight”. Moreover, since our mind is being constantly defiled, one cannot preserve the initial purity without the assiduous and continuous efforts of the watchful mind. We have to “wipe hour and hour” until our mind is cleansed of all defilements to reach Nirvana. Therefore, it is a tireless process to attain enlightenment, and this is why Shenxiu’s verse supports the claim that the Northern School teaches gradual enlightenment.

Huineng’s poem says (Mou-Lam and Price, p.74):

There is no bodhi tree,

Nor stand of a mirror bright.

Since all is void,

Where can the dust alight?

Huineng’s verse, on the other hand, contradicts sharply with Shenxiu’s teaching of the mind. Huineng believes that there is no duality between the pure mind and the defiled mind. By taking “All is void” as the fundamental assumption of their teaching, the Southern School rejects the existence of a mind as an entity that needs to be preserved. Since there is no mirror to be attached to, how can any “dust” alight in the first place? In other words, since there is no mind, there can be no defilement.

Instead of using the metaphor of mirror and dust, Huineng claims that the state of mind of an unenlightened individual is “the same way as the sun may be thickly veiled by cloud and unable to show his light.” (Mou-Lam and Price, p.85) This contrasts with Shenxiu’s metaphor as even if one cannot see the sun, it is still bright and clear. The clouds do not affect the brilliant nature of the sun, in the same way as erroneous views and false conceptions do not change the pure nature of our mind. Based on this metaphor, I present what I take to be the strongest argument put forward by Huineng simply as follows: instead of “beholding the mind”, one should employ inborn wisdom to realize his/her essential nature. The most crucial distinction between “mind” and “nature” is that we can never lose our nature, but we might lose our mind. “Nature” is essential, meaning that without which no existence can be possible. One should take note that although Huineng still uses the term “mind” on many occasions, he often interprets “mind” in the way that “nature” is normally understood. For example, Huineng says in the platform sutra, “our essence of mind is intrinsically pure” (Mou-Lam and Price, p.86). In other words, our mind (nature) is originally pure and clean and it remains pure and clean at all times. Huineng further associates the intrinsic mind with Buddha-nature by saying “Clear, free, empty, and silent, perception and action equally enlightened, mirrorlike awareness unobstructed – this is truly the inherent Buddha-nature” (Cleary, p.108). Therefore, our intrinsic mind is also our Buddha nature, which implies that all of us can become Buddhas. If one can see this point, one immediately gains enlightenment. Since all we have to do is to see our own mind, be our own teacher, and realize our own Buddhahood, the enlightenment process is innate, sudden, and spontaneous, as stated in the Platform Sutra, “A foolish passing thought makes one an ordinary man, while an enlightened second thought makes one a Buddha” (Mou-Lam and Price, p.83). As a result, one may say that the Southern School teaches sudden enlightenment.

All in all, it could be seen that the Northern School of Chan emphasizes self-purification, which is an accumulated effort. Hence, enlightenment has to be a gradual process. While for the Southern School of Chan, one has to employ inborn wisdom to realize his/her essential nature.

Despite finding arguments from both the Northern and Southern Schools compelling and sound, I am personally more inclined to the teachings of the Southern School for their close connections with my favorite branch of Chinese Philosophy – Daoism. For example, Zhuangzi takes Dao to be universal and permanent and claims that our ability to perceive Dao is innate, which is similar to Southern School’s idea of realizing the intrinsic mind and buddha-nature. In practice, Daoism also emphasizes the importance of following our intuitions instead of obeying a particular sutra or authority. I believe the similarities and differences between Daoism and Chan would be an interesting topic for future explorations.

Works Cited

Shenxiu. Guan Xin Lun.

Wong Mou-Lam, A. F. (2005). The Diamond Sutra and the Sutra of Hui-nen. N/A: Shambhala Publication.

Cleary, T. (1998). The Sutra of Hui-neng, Grand Master of Zen: With Hui-neng’s Commentary on the Diamond Sutra. N/A: Shambhala.

Exegetical Writing 2: The Analects – 4:10, 4:11, 14:22

September 28, 2021January 22, 2022Michael LiuLeave a comment

Translated Text

[4:10] The Master said: “When the noble man deals with the world he is not prejudiced for or against anything. He does what is Right.

[4:11] The Master said: “The superior man thinks of virtue; the inferior man thinks of possessions. The superior man seeks discipline; the inferior man seeks favors.”

[14:22] Zi Lu asked how to deal with a ruler. Confucius said, “Do not impose on him, and, moreover, withstand him to his face.”

Analysis

In passage 4:11, Confucius draws a clear distinction between a superior man and an inferior man. Confucius says, “The superior man thinks of virtue; the inferior man thinks of possessions. The superior man seeks discipline; the inferior man seeks favors.” From a general reading of the passage, it can be seen that a superior man is a morally exemplary person who possesses virtue and acts in accordance with moral principles, whereas an inferior man disregards virtues and focuses on petty affairs such as benefits and reputations. However, at a deeper level, it is crucial to understand how such a distinction is established.

A close reading of the passage 4:11 would require an examination of the verbs’ think’ and ‘seek.’ Both verbs, though appear to mean differently, are translated from the same Chinese character ‘Huai 怀.’ In common usage, ‘Huai’ means having an attitude or an emotion toward something or someone. For example, one may possess (Huai) a sense of respect toward her teacher. ‘Huai’ also implies that attitude can influence action, just as a person who is respectful to his teacher would go to class on time, participate in class, etc. In the context of Passage 4:11, what Confucius implies by ‘Huai’ is that a person habituates moral concerns with which she would carry out her moral judgments and act accordingly. In Great Learning, this idea is described as “What truly is within will be manifested without.” While the common conception of ‘think’ may allow assorted considerations, ‘Huai’ demands one’s devotion to a particular moral position. Perhaps, it would be better to say, “The superior man [cherishes] virtue.”

Passage 14:22 could serve as a further illustration of the idea of ‘Huai’ or ‘cherish.’ When Confucius’ student Zi Lu asks how a ruler should be served, Confucius replies, “Do not impose on him, and, moreover, withstand him to his face.” An explanation of this passage would be that an inferior man would avoid criticizing and upsetting the ruler. Instead, he flatters the ruler to gain favors and rewards for himself. On the other hand, a superior man who cherishes virtue would restrain himself from personal desires and preferences when dealing with public affairs as he is solely devoted to Confucius’s moral principles. He will be honest with the ruler and will stand up against the ruler’s wrong deeds.

Moreover, in passage 4:10, Confucius says, “A superior man in dealing with the world is not for anything or against anything. He follows righteousness as the standard.” One could treat this passage as a restatement of the arguments outlined in passage 4:11. The interpretation goes: a superior man in dealing with the (public) world is not for or against anyone’s personal benefits and preferences. He only makes moral judgments against the principle of righteousness (or virtue). In the context of this passage, righteousness carries a very similar meaning as ‘virtue’. This could be shown by passage 4:16, which is almost identical to passage 4:11, where Confucius says, “The superior man is versed in what is righteous. The inferior man is versed in what is profitable.”

To sum up, an inferior man would take personal benefits and preferences as his priority. His judgments are not morally guided, and his actions do not incorporate virtue. Conversely, a superior man would take virtues as his standard. He bases his moral judgments solely on moral principles and restrains himself from personal desires. Therefore, his actions are always appropriate to the Way.

Original Text

[4:10] 子曰：「君子之於天下也，無適也，無莫也，義之與比。」

[4:11] 子曰：「君子懷德，小人懷土；君子懷刑，小人懷惠。」

[14:22] 子路問事君。子曰：「勿欺也，而犯之。」

Exegetical Writing 1: Daodejing – chapter 38

September 28, 2021January 21, 2022Michael LiuLeave a comment

Translated Text

The man of superior virtue is not (conscious of) his virtue,

And in this way, he really possesses virtue.

The man of inferior virtue never loses (sight of) his virtue,

And in this way he loses his virtue.

The man of superior virtue takes no action, but has no ulterior motive to do so.

The man of superior humanity takes action, but has no ulterior motive to do so.

The man of superior righteousness takes action, and has an ulterior motive to do so.

The man of superior propriety takes action,

And when people do not respond to it, he will stretch his arms and force it on them.

Therefore, only when Dao is lost does the doctrine of virtue arise.

When virtue is lost, only then does the doctrine of humanity arise.

When humanity is lost, only then does the doctrine of righteousness arise.

When righteousness is lost, only then does the doctrine of propriety arise.

Now, propriety is a superficial expression of loyalty and faithfulness, and the beginning of disorder.

Those who are the first to know have the flowers (appearance) of Dao but are the beginning of ignorance.

For this reason the great man dwells in the thick (substantial), and does not rest with the thin (superficial).

He dwells in the fruit (reality), and does not rest with the flower (appearance).

Therefore, he rejects the one, and accepts the other.

Analysis

In this chapter, Laozi discusses the ways in which moral principles influence the evolution of society. In particular, he argues that conscious moral education, based on the tenets of Confucianism such as Humanity (Ren), Righteousness (Yi), and Propriety (Li), causes the decline of Dao and the corruption of social order.

Chapter 38 begins with:

The man of superior virtue (De) is not (conscious of) his virtue (De),

And in this way he really possesses virtue.

The man of inferior virtue (De) never loses (sight of) his virtue (De),

And in this way he loses his virtue.

What does Laozi mean when he says a virtuous man is unaware of his own ‘virtue’? How can one make sense of such an abstruse statement? First of all, note that the word ‘virtue’ was directly translated from the Chinese character ‘De (德)’. However, from a linguistic perspective, there was a subtle difference (which has often been ignored) between ‘De‘ and virtue in ancient times. While ‘virtue’ means high moral standards, ‘De‘ was more a less a neutral term: it refers to moral standards that can be evaluated (as good or bad). In ancient Chinese society, moral standards are evaluated based on what individuals got for the society (wealth, possessions, reputations, etc.), which ultimately boils down to what individuals got for themselves since there was very little distinction between individuals and a collective at that time. This is why in the Chinese language, ‘De’ is sometimes used as a homonym (德=得)^[1].

We can now go back and unpack the text. It is clear that ‘the man of superior virtue’ refers to a ruler with high moral standards (and conversely, ‘the man of inferior virtue’ refers to a ruler who has low moral standards). The subsequent ‘De’ is used as a homonym, meaning ‘to get’. Therefore, a ruler who has high moral standards is not conscious of what he gets for himself, so he is indeed virtuous. However, a ruler who has low moral standards pays too much attention to what he gets for himself (instead of for his people), so he loses his status (of being virtuous).

As the distinction between a virtuous ruler and a non-virtuous ruler is recognized, the remainder of the chapter becomes rather straightforward. Next, Laozi says,

The man of superior virtue takes no action, but has no ulterior motive to do so.

The man of superior humanity takes action, but has no ulterior motive to do so.

The man of superior righteousness takes action, and has an ulterior motive to do so.

The man of superior propriety takes action,

And when people do not respond to it, he will stretch his arms and force it on them.

In this part of the chapter, Laozi argues that as civilization progresses, a ruler who adopts the Confucian moral teaching would no longer have his political principle based on high moral standards. He would rule in terms of Humanity first, then shift to Righteousness, and eventually to Propriety. As a result, his political system is in continuous moral degeneration.

Laozi argues that the ruler in a political system based on high moral standards takes no action and has no ulterior motive because he has no personal desire to interfere with social development. However, a political system based on Humanity requires the ruler to take action because it aims for moral cultivation. For example, the ruler may have to introduce policies such as three years of moaning period for the death of parents in order to ‘morally’ transform his subjects. However, such a ruler still places the benefits of his people as the priority, so he is said to have no ulterior motive. On the other hand, a political system based on Righteousness implies that the ruler would move on to further interfere with society by establishing more rules on behavior and conduct. For example, he would ask his subject to kneel in front of him to show respect. However, his ulterior motive might be to fulfill his vanity and personal desire. Eventually, the ruler becomes morally corrupted and starts to rule in terms of Propriety by imposing rigid laws and behavioral codes on his people in order to exploit them. As a result, he inevitably faces opposition from his people, so he would have to use forces and punishments to achieve his goal.

Next, Laozi summarises his argument:

Therefore, only when Dao is lost does the doctrine of virtue arise.

When virtue is lost, only then does the doctrine of humanity arise.

When humanity is lost, only then does the doctrine of righteousness arise.

When righteousness is lost, only then does the doctrine of propriety arise.

This part can be easily understood as a repetition of Laozi’s previous statements.

Last, Laozi says,

Now, propriety is a superficial expression of loyalty and faithfulness, and the beginning of disorder.

Those who are the first to know have the flowers (appearance) of Dao but are the beginning of ignorance.

For this reason the great man dwells in the thick (substantial), and does not rest with the thin (superficial).

He dwells in the fruit (reality), and does not rest with the flower (appearance).

‘Those who are the first to know’ is translated from ‘前识者’. In the context of the text, it means ‘the political system prior to the introduction of the later’, which refers to Humanity (Ren). So by the end of chapter 38, Laozi reaches an important conclusion: we need to recognize that although Humanity is similar to Virtue in terms of its form, it would inevitably lead to chaos of disorder. Therefore, the ruler should abandon Confucian teachings and adhere to the Daoist teaching of Virtue.

To sum up, in chapter 38 of Daodejing, Laozi argues that when the ruler rules in accordance with Dao, his original nature is unspoiled and he is removed from personal desires, so he is unconsciously good. The decline of virtues comes with the development of conscious moral knowledge, which leads to the moral deterioration of the ruler and eventually causes hypocrisy and chaos among the people.

Notes

[1] This can be verified using a Chinese dictionary.

Original Text

上德不德，是以有德。下德不失德，是以無德。

上德無為而無以為；下德為之而有以為；上仁為之而無以為。上義為之而有以為。上禮為之而莫之應，則攘臂而扔之。

故失道而後德，失德而後仁，失仁而後義，失義而後禮；夫禮者，忠信之薄而亂之首。前識者，道之華，而愚之始。

是以大丈夫處其厚，不居其薄，處其實，不居其華，故去彼取此。