## NP-complete problems and physical reality (2005)

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Venue: | ACM SIGACT News Complexity Theory Column, March. ECCC |

Citations: | 32 - 4 self |

### BibTeX

@ARTICLE{Aaronson05np-completeproblems,

author = {Scott Aaronson},

title = {NP-complete problems and physical reality},

journal = {ACM SIGACT News Complexity Theory Column, March. ECCC},

year = {2005},

pages = {05--026}

}

### Years of Citing Articles

### OpenURL

### Abstract

Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing. ” The section on soap bubbles even includes some “experimental ” results. While I do not believe that any of the proposals will let us solve NP-complete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics. 1

### Citations

10922 |
Computers and Intractability: A Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...0) is in the lower left corner, and (1,1) in the upper right). There are two Steiner vertices, at roughly (.24,.19) and (.80,.26). a pretty sophisticated idea of why we have no idea [62]. See [69] or =-=[40]-=- for more information. Of course, even if there is no deterministic algorithm to solve NP-complete problems in polynomial time, there might be a probabilistic algorithm, or a nonuniform algorithm (one... |

1368 |
Quantum Computation and Quantum Information
- Nielsen, Chuang
- 2000
(Show Context)
Citation Context ...ism commensurate with the difficulty of NP-complete problems. For in quantum mechanics, we need a vector of 2 n complex numbers called “amplitudes” just to specify the state of an n-bit computer (see =-=[2, 36, 58]-=- for more details). Surely we could exploit this exponentiality inherent in Nature to try out all 2 n possible solutions to an NP-complete problem in parallel? Indeed, many popular articles on quantum... |

878 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
- Shor
- 1997
(Show Context)
Citation Context ...lity depending on its amplitude αx. The challenge is to arrange the computation in such a way that only the x’s we wish to see wind up with large values of αx. For the special case of factoring, Shor =-=[66]-=- showed that this could be done using a polynomial number of operations—but what about for NP-complete problems? The short answer is that we don’t know. Indeed, letting BQP be the class of problems so... |

843 | A fast quantum mechanical algorithm for database search
- Grover
- 1996
(Show Context)
Citation Context ...an oracle. In particular, they showed that any quantum algorithm that searches an unordered database of N items for a single “marked” item must query the database ∼ √ N times. (Soon afterward, Grover =-=[44]-=- showed that this is tight.) If we interpret the space of 2 n possible assignments to a Boolean formula ϕ as a “database,” and the satisfying assignments of ϕ as “marked items,” then Bennett et al.’s ... |

356 | On the power of quantum computation
- Simon
- 1994
(Show Context)
Citation Context ...bilities. First, many computer scientists see the study of “speculative” models of computation as at best a diversion from more serious work; this might explain why the groundbreaking papers of Simon =-=[67]-=- and Bennett et al. [17] were initially rejected from the major theory conferences. And second, many physicists see computational complexity as about as relevant to the mysteries of Nature as dentistr... |

348 |
Complexity and Real Computation
- Blum, Cucker, et al.
- 1997
(Show Context)
Citation Context ... floor function (or equivalently, to access a specific bit in a real number’s binary expansion) is essential here. If we drop that ability, then we obtain the beautiful theory of algebraic complexity =-=[18, 26]-=-, which has its own “P versus NP” questions over the real and complex numbers. These questions are logically unrelated to the original P versus NP question so far as anyone knows—possibly they are eas... |

313 | Strengths and weaknesses of quantum computing
- Bennett, Bernstein, et al.
- 1997
(Show Context)
Citation Context ...mputer scientists see the study of “speculative” models of computation as at best a diversion from more serious work; this might explain why the groundbreaking papers of Simon [67] and Bennett et al. =-=[17]-=- were initially rejected from the major theory conferences. And second, many physicists see computational complexity as about as relevant to the mysteries of Nature as dentistry or tax law. Today, how... |

269 | Quantum lower bounds by polynomials
- Beals, Buhrman, et al.
- 1998
(Show Context)
Citation Context ...s queried with the smallest total probability, and show that the algorithm will need many queries to notice this change. By now, many other proofs have been discovered, including that of Beals et al. =-=[13]-=-, which represents an efficient quantum algorithm’s acceptance probability by a low-degree polynomial, and then shows that no such polynomial exists; and that of Ambainis [9], which upper-bounds how m... |

246 |
A suggested interpretation of the quantum theory in terms of “hidden” variables
- Bohm
- 1952
(Show Context)
Citation Context ...anics with the “actual” values of certain observables (such as particle positions or momenta), together with rules for how those observables evolve in time. The most famous such theory is due to Bohm =-=[20]-=-, but there are many alternatives that are equally compatible with experiment. Indeed, a key feature of hidden-variable theories is that they reproduce the usual quantum-mechanical probabilities at an... |

200 |
Average case complete problems
- Levin
- 1986
(Show Context)
Citation Context ...lems are intractable in the physical world—eventually be seen as a principle of physics? In my view, the 18 The same question is also asked in the much more prosaic setting of average-case complexity =-=[55]-=-. 17sanswer ought to depend on (1) whether is there good evidence for the assumption, and (2) whether accepting it places interesting constraints on new physical theories. Regarding (1), we have seen ... |

182 |
Algebraic Complexity Theory
- Bürgisser, Clausen, et al.
- 1997
(Show Context)
Citation Context ... floor function (or equivalently, to access a specific bit in a real number’s binary expansion) is essential here. If we drop that ability, then we obtain the beautiful theory of algebraic complexity =-=[18, 26]-=-, which has its own “P versus NP” questions over the real and complex numbers. These questions are logically unrelated to the original P versus NP question so far as anyone knows—possibly they are eas... |

176 | Natural proofs
- Razborov, Rudich
- 1997
(Show Context)
Citation Context ...19,.06) (where (0,0) is in the lower left corner, and (1,1) in the upper right). There are two Steiner vertices, at roughly (.24,.19) and (.80,.26). a pretty sophisticated idea of why we have no idea =-=[62]-=-. See [69] or [40] for more information. Of course, even if there is no deterministic algorithm to solve NP-complete problems in polynomial time, there might be a probabilistic algorithm, or a nonunif... |

145 | Quantum lower bounds by quantum arguments
- Ambainis
- 2000
(Show Context)
Citation Context ...ing that of Beals et al. [13], which represents an efficient quantum algorithm’s acceptance probability by a low-degree polynomial, and then shows that no such polynomial exists; and that of Ambainis =-=[9]-=-, which upper-bounds how much the entanglement between the algorithm and database can increase via a single query. Both techniques have also led to lower bounds for many other problems besides databas... |

119 |
Does co-NP have short interactive proofs
- Boppana, H˚astad, et al.
- 1987
(Show Context)
Citation Context ...e problems for which no efficient quantum algorithm is known. On the other hand, SZK is not thought to contain the NP-complete problems; indeed, if it did then the polynomial hierarchy would collapse =-=[21]-=-. And it turns out that, even if we posit the unphysical ability to sample histories, we still could not solve NP-complete problems efficiently in the black-box setting! The best we could do is search... |

109 | Angular momentum: an approach to combinatorial spacetime
- Penrose
- 1971
(Show Context)
Citation Context ... Barrett-Crane model, this amplitude equals the product, over all Pachner moves in F, of an expression called a “10j symbol,” which can be evaluated according to rules originally developed by Penrose =-=[59]-=-. Complicated, perhaps, but this seems like the stuff out of which a computational model could be made. So two years ago I spoke with Dan Christensen, a mathematician who along with Greg Egan gave an ... |

107 |
Quantum lower bounds for the collision and the element distinctness problems
- Aaronson, Shi
(Show Context)
Citation Context ...〉 |σ〉 + |1〉 |τ〉) / √ 2 for some σ �= τ if they are isomorphic. Unfortunately, if then we measured this state in the standard basis, we would get no information whatsoever, and work of myself [1], Shi =-=[65]-=-, and Midrijanis [57] shows that no black-box quantum algorithm can do much better. But if only we could make a few “non-collapsing” measurements! Then we would see the same permutation each time in t... |

87 | PP is closed under intersection
- Beigel, Reingold, et al.
- 1991
(Show Context)
Citation Context ...tselect on the measurement outcome being |1〉. I then showed that PostBQP = PP, and used this fact to give a simple, quantum computing based proof of Beigel, Reingold, and Spielman’s celebrated result =-=[15]-=- that PP is closed under intersection. 10 Discussion Many of the deepest principles in physics are impossibility statements: for example, no superluminal signalling and no perpetual motion machines. W... |

87 | A modular functor which is universal for quantum computation
- Freedman, Larsen, et al.
(Show Context)
Citation Context ...ound. 7 Quantum Gravity Here we enter a realm of dragons, where speculation abounds but concrete ideas about computation are elusive. The one clear result is due to Freedman, Kitaev, Larsen, and Wang =-=[37, 38]-=-, who studied topological quantum field theories (TQFT’s). These theories, which arose from the work of Witten and others in the 1980’s, involve 2 spatial dimensions and 1 time dimension. Dropping fro... |

85 | Some NP-complete geometric problems
- Garey, Graham, et al.
- 1976
(Show Context)
Citation Context ...collection of line segments of minimum total length connecting the points, where the segments can meet at vertices (called Steiner vertices) other than the pegs themselves. Garey, Graham, and Johnson =-=[39]-=- showed that finding such a tree is NP-hard. 1 Yet a well-known piece of computer science folklore maintains that, if two glass plates with pegs between them are dipped into soapy water, then the soap... |

71 | The holographic principle
- Bousso
(Show Context)
Citation Context ...ic bound: the information content of any region is at most proportional to the surface area of the region, at a rate of one bit per Planck length squared, or 1.4 × 10 69 bits per square meter. Bousso =-=[23]-=-, whose survey paper on this subject is well worth reading by computer scientists, has reformulated the holographic bound in a generally covariant way, and marshaled a surprising amount of evidence fo... |

65 |
Turing machines that take advice
- Karp, Lipton
- 1982
(Show Context)
Citation Context ...ic algorithm to solve NP-complete problems in polynomial time, there might be a probabilistic algorithm, or a nonuniform algorithm (one that is different for each input length n). But Karp and Lipton =-=[53]-=- showed that either of these would have a consequence, namely the collapse of the polynomial hierarchy, that seems almost as implausible as P = NP. Also, Impagliazzo and Wigderson [50] gave strong evi... |

60 | Quantum algorithm for hilbert’s tenth problem
- Kieu
(Show Context)
Citation Context ...e problems in the physical world. Two examples of omissions are the gear-based computers of Vergis, Steiglitz, Dickinson [74], and the proposed adiabatic algorithm for the halting problem due to Kieu =-=[54]-=-. Also, I generally ignored papers about “hypercomputation” that did not try to forge some link, however tenuous, with the laws of physics as we currently understand them. 2 The Basics I will not say ... |

59 | Quantum lower bound for the collision problem
- Aaronson
- 2002
(Show Context)
Citation Context ...c, or (|0〉 |σ〉 + |1〉 |τ〉) / √ 2 for some σ �= τ if they are isomorphic. Unfortunately, if then we measured this state in the standard basis, we would get no information whatsoever, and work of myself =-=[1]-=-, Shi [65], and Midrijanis [57] shows that no black-box quantum algorithm can do much better. But if only we could make a few “non-collapsing” measurements! Then we would see the same permutation each... |

57 | A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem
- Farhi, Goldstone, et al.
(Show Context)
Citation Context ...all, NPcomplete problems are not black boxes, and classical algorithms such as backtrack search do exploit their structure. Why couldn’t a quantum algorithm do the same? A few years ago, Farhi et al. =-=[33]-=- announced a new quantum adiabatic algorithm, which can be seen as a quantum analogue of simulated annealing. Their algorithm is easiest to describe in a continuous-time setting, using the concepts of... |

51 |
Relativizations of the P=?NP question
- Baker, Gill, et al.
- 1975
(Show Context)
Citation Context ...e strong evidence that P = BPP; that is, that any probabilistic algorithm can be simulated by a deterministic one with polynomial slowdown. It is known that P �= NP in a “black box” or oracle setting =-=[11]-=-. This just means that any efficient algorithm for an NP-complete problem would have to exploit the problem’s structure in a nontrivial way, as opposed to just trying one candidate solution after anot... |

51 | The history and status of the P versus NP question
- Sipser
- 1992
(Show Context)
Citation Context ...here (0,0) is in the lower left corner, and (1,1) in the upper right). There are two Steiner vertices, at roughly (.24,.19) and (.80,.26). a pretty sophisticated idea of why we have no idea [62]. See =-=[69]-=- or [40] for more information. Of course, even if there is no deterministic algorithm to solve NP-complete problems in polynomial time, there might be a probabilistic algorithm, or a nonuniform algori... |

49 | Limitations of quantum advice and one-way communication
- Aaronson
(Show Context)
Citation Context ...ems solvable in quantum polynomial time, given a polynomial-size “quantum advice state” |ψn〉 that depends only on the input length n. Then recently I showed that NP �⊂ BQP/qpoly relative to an oracle =-=[3]-=-. Intuitively, even if the state |ψn〉 encoded the solutions to every 3SAT instance of size n, only a miniscule fraction of that information could be extracted by measuring |ψn〉, at least within the bl... |

48 |
A universal upper bound on the entropy to energy ratio for bounded systems
- Bekenstein
- 1981
(Show Context)
Citation Context ...sational partner is the one who would have to be placed in a spaceship. 10srelativity breaks down on length scales of order ℓP, and must be replaced by a quantum theory of gravity. Indeed, Bekenstein =-=[16]-=- gave an upper bound on the total information content of any isolated, weakly gravitating physical system, by assuming the Second Law of Thermodynamics and then considering a thought experiment in whi... |

46 |
What Is Thought
- Baum
- 2004
(Show Context)
Citation Context ...ly related to approximating the Kolmogorov complexity of a string or the circuit complexity of a Boolean function [8, 52]. 19sand that which we can understand we can predict. Indeed, in a recent book =-=[12]-=-, Eric Baum argues that much of what we call ‘insight’ or ‘intelligence’ simply means finding succinct representations for our sense data. On his view, the human mind is largely a bundle of hacks and ... |

45 | On the quantum algorithm for approximating the Jones polynomial. Unpublished
- Aharonov, Jones, et al.
- 2005
(Show Context)
Citation Context ... simulated in BQP, and some TQFT’s are universal for BQP. Unfortunately, the original papers on this discovery are all but impossible for a computer scientist to read, but Aharonov, Jones, and Landau =-=[7]-=- are currently working on a simplified presentation. From what I understand, it remains open to analyze the computational complexity of (3 + 1)dimensional quantum field theories even in flat spacetime... |

43 | How powerful is adiabatic quantum computation
- Dam, Mosca, et al.
- 2001
(Show Context)
Citation Context ...batic algorithm’s performance would be much more convincing if the known classical algorithms took exponential time on the same random instances. On the theoretical side, van Dam, Mosca, and Vazirani =-=[29]-=- constructed 3SAT instances for which the adiabatic algorithm provably takes exponential time, at least when the transition between the initial and final Hamiltonians is linear. Their instances involv... |

42 |
Precision tests of quantum mechanics
- Weinberg
- 1989
(Show Context)
Citation Context ...er than BQP. For, as we saw in Section 4, the linearity of quantum mechanics is what prevents one needle in an exponentially large haystack from shouting above the others. And as observed by Weinberg =-=[76]-=-, it seems difficult to change quantum mechanics in any consistent way while preserving linearity. But how drastic could the consequences possibly be, if we added a tiny nonlinear term to the Schrödin... |

39 | Quantum computing, postselection, and probabilistic polynomial-time
- Aaronson
- 2004
(Show Context)
Citation Context ...defined by Han, Hemaspaandra, and Thierauf [46] and which sits somewhere between MA and BPP NP . The exact power of BPPpath relative to more standard classes is still unknown. Also, in a recent paper =-=[5]-=- I defined a quantum analogue of BPPpath called PostBQP. This class consists of all problems solvable in quantum polynomial time, given the ability to measure a qubit with a nonzero probability of bei... |

39 | P=BPP unless E has subexponential circuits: derandomizing the XOR Lemma
- Impagliazzo, Wigderson
- 1997
(Show Context)
Citation Context ...Karp and Lipton [53] showed that either of these would have a consequence, namely the collapse of the polynomial hierarchy, that seems almost as implausible as P = NP. Also, Impagliazzo and Wigderson =-=[50]-=- gave strong evidence that P = BPP; that is, that any probabilistic algorithm can be simulated by a deterministic one with polynomial slowdown. It is known that P �= NP in a “black box” or oracle sett... |

38 | The complexity of analog computation
- Vergis, Steiglitz, et al.
- 1986
(Show Context)
Citation Context ...s, I was unable to discuss every paper related to the solvability of NP-complete problems in the physical world. Two examples of omissions are the gear-based computers of Vergis, Steiglitz, Dickinson =-=[74]-=-, and the proposed adiabatic algorithm for the halting problem due to Kieu [54]. Also, I generally ignored papers about “hypercomputation” that did not try to forge some link, however tenuous, with th... |

37 | Threshold computation and cryptographic security
- Han, Hemaspaandra, et al.
- 1997
(Show Context)
Citation Context ...sing anthropic postselection, could we do even more? Classically, it turns out that we could solve exactly the problems in a class called BPPpath, which was defined by Han, Hemaspaandra, and Thierauf =-=[46]-=- and which sits somewhere between MA and BPP NP . The exact power of BPPpath relative to more standard classes is still unknown. Also, in a recent paper [5] I defined a quantum analogue of BPPpath cal... |

36 |
Nonlinear quantum mechanics implies polynomial-time solution for NPcomplete and #P
- Abrams, Lloyd
- 1998
(Show Context)
Citation Context ...rs, Gisin [42] and Polchinski [60] showed that in most nonlinear variants of quantum mechanics, one could use entangled states to transmit superluminal signals. More relevant for us, Abrams and Lloyd =-=[6]-=- showed that one could solve NP-complete and even #P-complete problems in polynomial time—at least if the computation were error-free. Let us see why this is, starting with NP. Given a black-box funct... |

32 | Circuit minimization problem
- Kabanets, Cai
- 2000
(Show Context)
Citation Context ...hether they are complete under sufficiently strong reductions. Both problems are closely related to approximating the Kolmogorov complexity of a string or the circuit complexity of a Boolean function =-=[8, 52]-=-. 19sand that which we can understand we can predict. Indeed, in a recent book [12], Eric Baum argues that much of what we call ‘insight’ or ‘intelligence’ simply means finding succinct representation... |

30 |
Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik, 37:863–867
- Born
- 1926
(Show Context)
Citation Context ...ssues such as these are still not settled. 14 In the early days of quantum mechanics, there was much confusion about the operational meaning of the wavefunction. (Even in Born’s celebrated 1926 paper =-=[22]-=-, the idea that one has to square amplitudes to get probabilities only appeared in a footnote added in press!) Similarly, Einstein struggled for years to extract testable physics from a theory in whic... |

29 | The black hole final state
- Horowitz, Maldacena
(Show Context)
Citation Context ...bjection is that, even if the NP Hardness Assumption can be formulated precisely, it is unlike any other physical principle we know. How could a statement that refers not to 19 Horowitz and Maldacena =-=[49]-=- recently proposed such a modification as a way to resolve the black hole information loss paradox. See also a comment by Gottesman and Preskill [43]. 20 Of course, your “free will” to choose an input... |

25 |
The End of Science
- Horgan
- 1997
(Show Context)
Citation Context ...needle in the mathematical haystack.” —From Quarantine [31], a 1992 science-fiction novel by Greg Egan If I had to debate the science writer John Horgan’s claim that basic science is coming to an end =-=[48]-=-, my argument would lean heavily on one fact: it has been only a decade since we learned that quantum computers could factor integers in polynomial time. In my (unbiased) opinion, the showdown that qu... |

24 |
On the power of random access machines
- Schönhage
- 1979
(Show Context)
Citation Context ...ting. (For present purposes, an analog computer is a machine that performs a discrete sequence of steps, but on unlimited-precision real numbers.) As an example of such an approach, in 1979 Schönhage =-=[64]-=- showed how to solve NP-complete and even PSPACE-complete problems in polynomial time, given the ability to compute x + y, x − y, xy, x/y, and ⌊x⌋ in a single time step for any two real numbers x and ... |

23 |
Quantum mechanics near closed timelike lines
- Deutsch
- 1991
(Show Context)
Citation Context ...mpany that will invent the plexiglass, and reveals its molecular formula to that company. The question is, where did the work of inventing the formula take place? In a classic paper on CTC’s, Deutsch =-=[30]-=- observes that, in contrast to the much better-known grandfather paradox, the “knowledge creation paradox” involves no logical contradiction. The only paradox is a complexity-theoretic one: a difficul... |

23 | One complexity theorist’s view of quantum computing
- Fortnow
(Show Context)
Citation Context ...ism commensurate with the difficulty of NP-complete problems. For in quantum mechanics, we need a vector of 2 n complex numbers called “amplitudes” just to specify the state of an n-bit computer (see =-=[2, 36, 58]-=- for more details). Surely we could exploit this exponentiality inherent in Nature to try out all 2 n possible solutions to an NP-complete problem in parallel? Indeed, many popular articles on quantum... |

22 |
The computational complexity of the Tutte plane: the bipartite case
- Vertigan, Welsh
- 1992
(Show Context)
Citation Context ...) using Penrose’s rules is #P-complete. This follows by a simple reduction from counting the number of edge 3-colorings of a trivalent planar graph, which was proven #P-complete by Vertigan and Welsh =-=[75]-=-. But what about simulating the dynamics of (say) the Barrett-Crane model? Here we quickly ran into problems: for example, in summing over all spin foams between two spin networks, should one impose a... |

20 |
Weinberg’s non-linear quantum mechanics and superluminal communications
- Gisin
- 1990
(Show Context)
Citation Context ...g linearity. But how drastic could the consequences possibly be, if we added a tiny nonlinear term to the Schrödinger equation (which describes how quantum states evolve in time)? For starters, Gisin =-=[42]-=- and Polchinski [60] showed that in most nonlinear variants of quantum mechanics, one could use entangled states to transmit superluminal signals. More relevant for us, Abrams and Lloyd [6] showed tha... |

17 |
Weinberg’s nonlinear quantum mechanics and the Einstein-Podolsky-Rosen paradox
- Polchinski
- 1991
(Show Context)
Citation Context ... drastic could the consequences possibly be, if we added a tiny nonlinear term to the Schrödinger equation (which describes how quantum states evolve in time)? For starters, Gisin [42] and Polchinski =-=[60]-=- showed that in most nonlinear variants of quantum mechanics, one could use entangled states to transmit superluminal signals. More relevant for us, Abrams and Lloyd [6] showed that one could solve NP... |

16 | What can be efficiently reduced to the Kolmogorov-random strings
- Allender, Buhrman, et al.
- 2004
(Show Context)
Citation Context ...hether they are complete under sufficiently strong reductions. Both problems are closely related to approximating the Kolmogorov complexity of a string or the circuit complexity of a Boolean function =-=[8, 52]-=-. 19sand that which we can understand we can predict. Indeed, in a recent book [12], Eric Baum argues that much of what we call ‘insight’ or ‘intelligence’ simply means finding succinct representation... |

16 |
The quantum adiabatic optimization algorithm and local minima
- Reichardt
- 2004
(Show Context)
Citation Context ... the adiabatic algorithm computes not merely whether X satisfies ϕ, but also how many clauses it satisfies. And this information turns out to be sufficient to reconstruct ϕ itself. Recently Reichardt =-=[63]-=-, building on work of Farhi, Goldstone, and Gutmann [32], has constructed 3SAT instances for which the adiabatic algorithm takes polynomial time, whereas simulated annealing takes exponential time. Th... |

16 |
On the Pilot-Wave Theory of Classical
- Valentini
- 1992
(Show Context)
Citation Context ...to ∼ N 1/2 with Grover’s algorithm. But even if a hidden-variable picture is correct, are these considerations relevant to any computations we could perform? They would be, if a proposal of Valentini =-=[73, 72]-=- were to pan out. Valentini argues that the |ψ| 2 probability law merely reflects a statistical equilibrium (analogous to thermal equilibrium), and that it might be possible to find “nonequilibrium ma... |