- Email: [email protected]
Infinite networks: II—Resistance in an infinite grid
JOURNAL
OF MATHEMATICAL
infinite
Networks
ANALYSIS
AND
40, 30-35 (1972)
APPLICATIONS
:Il-Resistance
in an Infinite
Grid*
HARLEY FLANDERS Department of Mathematics, Purdue University, Lafayette, Indiana 46207 and Tel Aviv University, Ramat Aviv, Israel Submitted by R. J. Dufin
Consider the infinite grid in the plane consistingof all horizontal and all vertical segmentsjoining lattice points (points with integer coordinates). Supposeeach of these segmentsis a one-ohm resistor. What is the effective resistance(driving-point impedance)between a given pair of lattice points? This problem appearsseveraltimes in the literature. Let us examine what hasbeen said.St6hr [12] mentionsthe problem (in somewhatgreater generality) at the beginning of his three-part article, but never returns to it. Purcell [lo] posesthe problem of showing that the effective resistanceacross adjacent nodes is 8 ohm, and assertsthat this can be found by symmetry and superposition alone. Foster [8] discussesthe problem for adjacent nodes (in this and someother plane configurations) and finds the answer$ ohm by assumingthat his averaging formula for finite networks remainsvalid in the limit. His argument is repeated by Weinberg [14], who also gives another derivation of 4 ohm by a symmetry argument. (I cannot follow the symmetry argument, becausehe connectsan external current generator which seemsto destroy the symmetry.) Finally, Van der Pol [131assertsthat the fundamental solution of the harmonic difference equation solvesthe problem in general and gives a closedform of this fundamental solution (ashave earlier writers). In particular, he finds the effective resistancebetween (0,O) and (n, n) to be
(All effective resistancescan be derived easily from these.) Incidentally, fundamental work on the harmonic difference equation, particularly on the Dirichlet problem, is contained in Courant [l] and Courant, Friedrichs, and Lewy [2]. According to the calculations in Stohr [12], Spitzer [l 11,Van der Pol [13], * This work was partially supported by an NSF Grant.
30 Copyright All rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
INFINITE
NETWORKS
31
II
and McCrea and Whipple [9], the fundamental solution (normalized to give the effective resistance) between (0,O) and (m, n) has several alternate expressions.Other expressionscan be found in Duffin [4, p. 3501and Duffin and Shaffer [5, p. 5931. u(m, n) = - &
j,
[(l -i)“‘”
(1 - g)“-”
(1 - [t)-n+n
. (1 - &)-m-n - 11 f for n 3 0, where 5 = esni/s, u(m, n) = - +
I
z7(cos me) ((2 - cos e) - [(2 - cos t9)2[(2 - cose>2- 111/s
1]1~2}~nf- 1dB
0
(Stohr),
(Spitzer), u(n.2,n) = &
1: [l - (z)“‘”
(s)‘“-“‘I
$ (Van der Pohl),
m1 - e-lmlo,cosnc9 4 sinh 01 0
u(m, n) = -J- J”
where cos0 + cashOL= 2 (McCrea, Whipple). Several values of u(m, n) are tabulated in Duffin and Shelly [6], McCrea and Whipple [9], and Stohr [12]. In Table I we summarize these results. Note that ~(m, n) = u(n, m) = u(- m, n) = etc. The missingentries can be computed by symmetry and the difference equation P, below. This fundamental solution u(m, n) is characterized by four properties: Pl
u(0, 0) = 0;
P2
u(m+l,n)+u(m-l,n)+u(m,n+1)+u(m,n-1)-44u(m,n) = 0, h 4 # (0, 0); (sameexpression) = 2, (m, 4 = (0, 0);
p3 P4
u(m, n) > 0 for all (m, n)
P,’
u(m, n) = o[(m2 + n2)1/2] at co
(Spitzer);
or
409/40/r-3
(Stohr).
-
\
?I
8
7
6
5
4
3
2
I
0
m
-
0
0
2
1
7r
z
1
3x
8
2-A
77
2
- 46 15rr
-4+E
2
---17
3
37
24
1057r
352
L+$
6-K
-T+3?r
49
40 -
7-r
236
3Tr
368
4
n
240
Some
TABLE
3157r
1126
++;
--8+g
--15 2
97
71
77
77
6646
3a
1880
5
I
1569
3465~
13016
336 -
-2
1042 -
of u(m, n)
2236
-140-t15x
2
401 --~
Values
~ 105n
110472
tx37020
15a
~49052
6
45045a
176138
-4376
11073 --2 1443874 + ___ 105lr
260848 15Tr
7
45045r
182144
29856
-
8
9848128 ___1057f
INFINITE
NETWORKS
33
II
I believe these results do indeed give the correct solution for the effective resistance, however, as far as I can see this has never been proved rigorously. First of all the assumption that a specific infinite network of resistors has a solution is premised on the existence of a theory of infinite networks. Until Flanders [7], this did not exist. Second, the solution given above must satisfy the requirements of this theory of infinite networks in order that it be the correct solution to the given grid problem. If B, , B, ,... denote the directed branches of an infinite network, if ri > 0 denotes the resistance of branch Bi , and if the network is energized by a finite number of sources, then it is shown in Flanders [7] that there is a unique flow of current I = C aiBi subject to Nl
Kirchhoff’s
node law (I is a cycle),
N2
Kirchhoff’s
loop law,
NS
C ai2ri < co,
JXi
I is the limit of finite cycles.
(One consequence of this result is that conservation of energy holds. This is Corollary 4 of the Existence theorem in [73].) Consider now the following problem, closely related to the effective resistance problem. Insert a one-volt source in the branch from (0,O) to (0, 1). What is the resulting current flow guaranteed by N,-N,? Let Z,, denote the unit square whose lower left-hand corner is (m, n), considered as a cycle with the clockwise-rotation sense. Write the solution in the form 1 = 1 Gmzrl, so that N, and N, are satisfied. The loop law N, is satisfied provided cm+1.n
+
Cm-1.72
-1
-1
+
Gn*n+1
+
Gw-1
1
cm,
4
=
@IO),
0
(m,
n)
#
(0,
(m, n) = (-
-
01,
(-
%m
1, o>,
1,O).
This suggests we try cm.,=g.(,,n)-&u(m+l,n), where ~(111,n) is the fundamental solution discussed above. The current in each branch is the difference of the “loop currents” from the two adjacent square loops, hence condition N, is equivalent to
34
FLANDERS
since each branch resistance is unity. If this is correct, an easy superposition argument shows that the effective resistance between (0,O) and (m, n) is precisely u(m, n). The crux of the matter then is to prove, for the fundamental solution u(m, n), that the second differences are square summable: C [u(m + 2,
n) - 2u(m+ 1, n) + u(m,n)l” < 00,
1 [u(m + 1, n + 1) - u(m + 1, n) - u(m, n + 1) + u(m, 72)]” < co. These sums indeed are finite, and that is a consequence of an asymptotic approximation for u(m, n). The approximations in Spitzer [l l] and McCrea and Whipple [9] are not sufficiently sharp for this purpose, but Stijhr gives the following result: u(m, n) = i
(3 In 2 + 2~) + + ln(m2 + 122)1/2+ 0 (A)
,
where y is the Euler c0nstant.l It is easy to see that any second difference of ln(m2 + n2)li2 is O(m2 + n2)-l and that
c
(m2
;
*2)2
<
co,
so square summability is assured and the given solution is valid. Finding explicit solutions for the other networks considered by Foster [8] and Weinberg [14] seems very difficult.2
ACKNOWLEDGMENT I am indebted
to Frank
Spitzer
for pointing
out Van der Pohl’s
and Stohr’s
work.
Congr.
Mat.
REFERENCES 1. R. COURANT, Uber partielle Bdogna 3 (1928), 83-89.
Differenzengleichungen,
Atti
Int.
i An even sharper asymptotic formula is given in Duffin and Shaffer [5, p. 5941, but Stijhr’s is adequate here. 2 The three-dimensional rectangular grid is discussed extensively in Duflin [3], where the application to resistance is noted and an asymptotic formula given, and in Duffin and Shelly [6], where numerical values are tabulated.
INFINITE
NETWORKS
II
35
2. R. COURANT, K. FRIEDRICHS, AND H. LEWY, Uber die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 (1928), 32-74. 3. R. J. DUPFIN, Discrete potential theory, Duke Math. J. 20 (1953), 233-251. 4. R. J. DUFFIN, Basic properties of discrete analytic functions, Duke Math. 1. 23 (1956), 335-364. 5. R. J. DUFFIN AND D. H. SHAFFFB, Asymptotic expansion of double Fourier transforms, Duke Math. J. 27 (1960), 581-596. 6. R. J. DUFFIN AND E. P. SHELLY, Difference equations of polyharmonic type, Duke Math. J. 25 (1958), 209-238. 7. H. FLANDERS, Infinite networks: I-Resistive networks, IEEE Tram. Cirmit Theory CT-18 (1971), 326-331. 8. R. M. FOSTER, The average impedance of an electric network, in “Contributions to Applied Mechanics,” Reisner Anniversary Volume,” pp. 333-340, Edwards Bros., Ann Arbor, MI, 1949. 9. W. H. MCCREA AND F. J. W. WHIPPLE, Random paths in two and three dimensions, Proc. Roy. Sot. Edinburgh 60 (1940), 281-298 (esp. 287-294). 10. E. M. PURCELL, “Electricity and Magnetism,” p. 424, McGraw-Hill, New York, 1963. 11. F. SPITZER, “Principles of Random Walk,” pp. 124, 148-151, Van Nostrand, Princeton, NJ, 1964. 12. A. ST~HR, Uber einige lineare partielle Differenzengleichungen mit konstanten Koeffizienten I, III, M&h. AJuch. 3 (1950), 208-242 (esp. 214-215), and 330-357 (esp. 342). 13. B. VAN DER POL, The finite difference analogy of the periodic wave equation and the potential equation, Appendix IV in M. KAC, “Probability and Related Topics in Physical Sciences,” pp. 237-257, Interscience, London, 1959. 14. L. WEINBERG, “Network Analysis and Synthesis,” pp. 171-176, McGraw-Hill, New York, 1962.
OF MATHEMATICAL
infinite
Networks
ANALYSIS
AND
40, 30-35 (1972)
APPLICATIONS
:Il-Resistance
in an Infinite
Grid*
HARLEY FLANDERS Department of Mathematics, Purdue University, Lafayette, Indiana 46207 and Tel Aviv University, Ramat Aviv, Israel Submitted by R. J. Dufin
Consider the infinite grid in the plane consistingof all horizontal and all vertical segmentsjoining lattice points (points with integer coordinates). Supposeeach of these segmentsis a one-ohm resistor. What is the effective resistance(driving-point impedance)between a given pair of lattice points? This problem appearsseveraltimes in the literature. Let us examine what hasbeen said.St6hr [12] mentionsthe problem (in somewhatgreater generality) at the beginning of his three-part article, but never returns to it. Purcell [lo] posesthe problem of showing that the effective resistanceacross adjacent nodes is 8 ohm, and assertsthat this can be found by symmetry and superposition alone. Foster [8] discussesthe problem for adjacent nodes (in this and someother plane configurations) and finds the answer$ ohm by assumingthat his averaging formula for finite networks remainsvalid in the limit. His argument is repeated by Weinberg [14], who also gives another derivation of 4 ohm by a symmetry argument. (I cannot follow the symmetry argument, becausehe connectsan external current generator which seemsto destroy the symmetry.) Finally, Van der Pol [131assertsthat the fundamental solution of the harmonic difference equation solvesthe problem in general and gives a closedform of this fundamental solution (ashave earlier writers). In particular, he finds the effective resistancebetween (0,O) and (n, n) to be
(All effective resistancescan be derived easily from these.) Incidentally, fundamental work on the harmonic difference equation, particularly on the Dirichlet problem, is contained in Courant [l] and Courant, Friedrichs, and Lewy [2]. According to the calculations in Stohr [12], Spitzer [l 11,Van der Pol [13], * This work was partially supported by an NSF Grant.
30 Copyright All rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
INFINITE
NETWORKS
31
II
and McCrea and Whipple [9], the fundamental solution (normalized to give the effective resistance) between (0,O) and (m, n) has several alternate expressions.Other expressionscan be found in Duffin [4, p. 3501and Duffin and Shaffer [5, p. 5931. u(m, n) = - &
j,
[(l -i)“‘”
(1 - g)“-”
(1 - [t)-n+n
. (1 - &)-m-n - 11 f for n 3 0, where 5 = esni/s, u(m, n) = - +
I
z7(cos me) ((2 - cos e) - [(2 - cos t9)2[(2 - cose>2- 111/s
1]1~2}~nf- 1dB
0
(Stohr),
(Spitzer), u(n.2,n) = &
1: [l - (z)“‘”
(s)‘“-“‘I
$ (Van der Pohl),
m1 - e-lmlo,cosnc9 4 sinh 01 0
u(m, n) = -J- J”
where cos0 + cashOL= 2 (McCrea, Whipple). Several values of u(m, n) are tabulated in Duffin and Shelly [6], McCrea and Whipple [9], and Stohr [12]. In Table I we summarize these results. Note that ~(m, n) = u(n, m) = u(- m, n) = etc. The missingentries can be computed by symmetry and the difference equation P, below. This fundamental solution u(m, n) is characterized by four properties: Pl
u(0, 0) = 0;
P2
u(m+l,n)+u(m-l,n)+u(m,n+1)+u(m,n-1)-44u(m,n) = 0, h 4 # (0, 0); (sameexpression) = 2, (m, 4 = (0, 0);
p3 P4
u(m, n) > 0 for all (m, n)
P,’
u(m, n) = o[(m2 + n2)1/2] at co
(Spitzer);
or
409/40/r-3
(Stohr).
-
\
?I
8
7
6
5
4
3
2
I
0
m
-
0
0
2
1
7r
z
1
3x
8
2-A
77
2
- 46 15rr
-4+E
2
---17
3
37
24
1057r
352
L+$
6-K
-T+3?r
49
40 -
7-r
236
3Tr
368
4
n
240
Some
TABLE
3157r
1126
++;
--8+g
--15 2
97
71
77
77
6646
3a
1880
5
I
1569
3465~
13016
336 -
-2
1042 -
of u(m, n)
2236
-140-t15x
2
401 --~
Values
~ 105n
110472
tx37020
15a
~49052
6
45045a
176138
-4376
11073 --2 1443874 + ___ 105lr
260848 15Tr
7
45045r
182144
29856
-
8
9848128 ___1057f
INFINITE
NETWORKS
33
II
I believe these results do indeed give the correct solution for the effective resistance, however, as far as I can see this has never been proved rigorously. First of all the assumption that a specific infinite network of resistors has a solution is premised on the existence of a theory of infinite networks. Until Flanders [7], this did not exist. Second, the solution given above must satisfy the requirements of this theory of infinite networks in order that it be the correct solution to the given grid problem. If B, , B, ,... denote the directed branches of an infinite network, if ri > 0 denotes the resistance of branch Bi , and if the network is energized by a finite number of sources, then it is shown in Flanders [7] that there is a unique flow of current I = C aiBi subject to Nl
Kirchhoff’s
node law (I is a cycle),
N2
Kirchhoff’s
loop law,
NS
C ai2ri < co,
JXi
I is the limit of finite cycles.
(One consequence of this result is that conservation of energy holds. This is Corollary 4 of the Existence theorem in [73].) Consider now the following problem, closely related to the effective resistance problem. Insert a one-volt source in the branch from (0,O) to (0, 1). What is the resulting current flow guaranteed by N,-N,? Let Z,, denote the unit square whose lower left-hand corner is (m, n), considered as a cycle with the clockwise-rotation sense. Write the solution in the form 1 = 1 Gmzrl, so that N, and N, are satisfied. The loop law N, is satisfied provided cm+1.n
+
Cm-1.72
-1
-1
+
Gn*n+1
+
Gw-1
1
cm,
4
=
@IO),
0
(m,
n)
#
(0,
(m, n) = (-
-
01,
(-
%m
1, o>,
1,O).
This suggests we try cm.,=g.(,,n)-&u(m+l,n), where ~(111,n) is the fundamental solution discussed above. The current in each branch is the difference of the “loop currents” from the two adjacent square loops, hence condition N, is equivalent to
34
FLANDERS
since each branch resistance is unity. If this is correct, an easy superposition argument shows that the effective resistance between (0,O) and (m, n) is precisely u(m, n). The crux of the matter then is to prove, for the fundamental solution u(m, n), that the second differences are square summable: C [u(m + 2,
n) - 2u(m+ 1, n) + u(m,n)l” < 00,
1 [u(m + 1, n + 1) - u(m + 1, n) - u(m, n + 1) + u(m, 72)]” < co. These sums indeed are finite, and that is a consequence of an asymptotic approximation for u(m, n). The approximations in Spitzer [l l] and McCrea and Whipple [9] are not sufficiently sharp for this purpose, but Stijhr gives the following result: u(m, n) = i
(3 In 2 + 2~) + + ln(m2 + 122)1/2+ 0 (A)
,
where y is the Euler c0nstant.l It is easy to see that any second difference of ln(m2 + n2)li2 is O(m2 + n2)-l and that
c
(m2
;
*2)2
<
co,
so square summability is assured and the given solution is valid. Finding explicit solutions for the other networks considered by Foster [8] and Weinberg [14] seems very difficult.2
ACKNOWLEDGMENT I am indebted
to Frank
Spitzer
for pointing
out Van der Pohl’s
and Stohr’s
work.
Congr.
Mat.
REFERENCES 1. R. COURANT, Uber partielle Bdogna 3 (1928), 83-89.
Differenzengleichungen,
Atti
Int.
i An even sharper asymptotic formula is given in Duffin and Shaffer [5, p. 5941, but Stijhr’s is adequate here. 2 The three-dimensional rectangular grid is discussed extensively in Duflin [3], where the application to resistance is noted and an asymptotic formula given, and in Duffin and Shelly [6], where numerical values are tabulated.
INFINITE
NETWORKS
II
35
2. R. COURANT, K. FRIEDRICHS, AND H. LEWY, Uber die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 (1928), 32-74. 3. R. J. DUPFIN, Discrete potential theory, Duke Math. J. 20 (1953), 233-251. 4. R. J. DUFFIN, Basic properties of discrete analytic functions, Duke Math. 1. 23 (1956), 335-364. 5. R. J. DUFFIN AND D. H. SHAFFFB, Asymptotic expansion of double Fourier transforms, Duke Math. J. 27 (1960), 581-596. 6. R. J. DUFFIN AND E. P. SHELLY, Difference equations of polyharmonic type, Duke Math. J. 25 (1958), 209-238. 7. H. FLANDERS, Infinite networks: I-Resistive networks, IEEE Tram. Cirmit Theory CT-18 (1971), 326-331. 8. R. M. FOSTER, The average impedance of an electric network, in “Contributions to Applied Mechanics,” Reisner Anniversary Volume,” pp. 333-340, Edwards Bros., Ann Arbor, MI, 1949. 9. W. H. MCCREA AND F. J. W. WHIPPLE, Random paths in two and three dimensions, Proc. Roy. Sot. Edinburgh 60 (1940), 281-298 (esp. 287-294). 10. E. M. PURCELL, “Electricity and Magnetism,” p. 424, McGraw-Hill, New York, 1963. 11. F. SPITZER, “Principles of Random Walk,” pp. 124, 148-151, Van Nostrand, Princeton, NJ, 1964. 12. A. ST~HR, Uber einige lineare partielle Differenzengleichungen mit konstanten Koeffizienten I, III, M&h. AJuch. 3 (1950), 208-242 (esp. 214-215), and 330-357 (esp. 342). 13. B. VAN DER POL, The finite difference analogy of the periodic wave equation and the potential equation, Appendix IV in M. KAC, “Probability and Related Topics in Physical Sciences,” pp. 237-257, Interscience, London, 1959. 14. L. WEINBERG, “Network Analysis and Synthesis,” pp. 171-176, McGraw-Hill, New York, 1962.