Misplaced Pages

#267732

65-504: (Redirected from Big Numbers ) Big numbers may refer to: Large numbers , numbers that are significantly larger than those ordinarily used in everyday life Arbitrary-precision arithmetic , also called bignum arithmetic Big Numbers (comics) , an unfinished comics series by Alan Moore and Bill Sienkiewicz See also [ edit ] Names of large numbers List of arbitrary-precision arithmetic software Topics referred to by

130-424: A < 10 {\displaystyle 1<a<10} ). (See also extension of tetration to real heights .) Thus googolplex is 10 10 100 = ( 10 ↑ ) 2 100 = ( 10 ↑ ) 3 2 {\displaystyle 10^{10^{100}}=(10\uparrow )^{2}100=(10\uparrow )^{3}2} . Another example: Thus the "order of magnitude" of

195-552: A = ( 10 ↑ ↑ ) 6 a {\displaystyle 10\uparrow (10\uparrow \uparrow )^{5}a=(10\uparrow \uparrow )^{6}a} , and 10 ↑ ( 10 ↑ ↑ ↑ 3 ) = 10 ↑ ↑ ( 10 ↑ ↑ 10 + 1 ) ≈ 10 ↑ ↑ ↑ 3 {\displaystyle 10\uparrow (10\uparrow \uparrow \uparrow 3)=10\uparrow \uparrow (10\uparrow \uparrow 10+1)\approx 10\uparrow \uparrow \uparrow 3} . Thus

260-402: A certain inflationary model with an inflaton whose mass is 10 Planck masses . This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is in a model where the universe's history repeats itself arbitrarily many times due to properties of statistical mechanics ; this is the time scale when it will first be somewhat similar (for

325-526: A certain amount of time. The result is obtained with the exact time reversible dynamical equations of motion and the universal causation proposition. The fluctuation theorem is obtained using the fact that dynamics is time reversible. Quantitative predictions of this theorem have been confirmed in laboratory experiments at the Australian National University conducted by Edith M. Sevick et al. using optical tweezers apparatus. This theorem

390-630: A generalized sense. A crude way of specifying how large a number is, is specifying between which two numbers in this sequence it is. More precisely, numbers in between can be expressed in the form ( 10 ↑ ) n a {\displaystyle (10\uparrow )^{n}a} , i.e., with a power tower of 10s, and a number at the top, possibly in scientific notation, e.g. 10 10 10 10 10 4.829 = ( 10 ↑ ) 5 4.829 {\displaystyle 10^{10^{10^{10^{10^{4.829}}}}}=(10\uparrow )^{5}4.829} ,

455-469: A more precise description of a number also specifies the value of this number between 1 and 10, or the previous number (taking the logarithm one time less) between 10 and 10 , or the next, between 0 and 1. Note that I.e., if a number x is too large for a representation ( 10 ↑ ) n x {\displaystyle (10\uparrow )^{n}x} the power tower can be made one higher, replacing x by log 10 x , or find x from

520-520: A new letter every time, e.g. as a subscript, such that there are numbers of the form f k m ( n ) {\displaystyle f_{k}^{m}(n)} where k and m are given exactly and n is an integer which may or may not be given exactly. Using k =1 for the f above, k =2 for g , etc., obtains (10→10→ n → k ) = f k ( n ) = f k − 1 n ( 1 ) {\displaystyle f_{k}(n)=f_{k-1}^{n}(1)} . If n

585-427: A number (like using the superscript of the arrow instead of writing many arrows). Introducing a function f ( n ) = 10 ↑ n 10 {\displaystyle f(n)=10\uparrow ^{n}10} = (10 → 10 → n ), these levels become functional powers of f , allowing us to write a number in the form f m ( n ) {\displaystyle f^{m}(n)} where m

650-488: A number (on a larger scale than usually meant), can be characterized by the number of times ( n ) one has to take the l o g 10 {\displaystyle log_{10}} to get a number between 1 and 10. Thus, the number is between 10 ↑ ↑ n {\displaystyle 10\uparrow \uparrow n} and 10 ↑ ↑ ( n + 1 ) {\displaystyle 10\uparrow \uparrow (n+1)} . As explained,

715-504: A number between 10 ↑ ↑ 5 {\displaystyle 10\uparrow \uparrow 5} and 10 ↑ ↑ 6 {\displaystyle 10\uparrow \uparrow 6} (note that 10 ↑ ↑ n < ( 10 ↑ ) n a < 10 ↑ ↑ ( n + 1 ) {\displaystyle 10\uparrow \uparrow n<(10\uparrow )^{n}a<10\uparrow \uparrow (n+1)} if 1 <

SECTION 10

#1732884403268

780-417: A number in the form g m ( n ) {\displaystyle g^{m}(n)} where m is given exactly and n is an integer which may or may not be given exactly. For example, if (10→10→ m →3) = g (1). If n is large any of the above can be used for expressing it. Similarly a function h , etc. can be introduced. If many such functions are required, they can be numbered instead of using

845-866: A number too large to write down in the Conway chained arrow notation it size can be described by the length of that chain, for example only using elements 10 in the chain; in other words, one could specify its position in the sequence 10, 10→10, 10→10→10, .. If even the position in the sequence is a large number same techniques can be applied again. Numbers expressible in decimal notation: Numbers expressible in scientific notation: Numbers expressible in (10 ↑) k notation: Bigger numbers: Some notations for extremely large numbers: These notations are essentially functions of integer variables, which increase very rapidly with those integers. Ever-faster-increasing functions can easily be constructed recursively by applying these functions with large integers as argument. A function with

910-435: A number with a sequence of powers ( 10 ↑ n ) k n {\displaystyle (10\uparrow ^{n})^{k_{n}}} with decreasing values of n (with exactly given integer exponents k n {\displaystyle {k_{n}}} ) with at the end a number in ordinary scientific notation. Whenever a k n {\displaystyle {k_{n}}}

975-446: A particular power or to adjust the value on which it act, instead it is possible to simply use a standard value at the right, say 10, and the expression reduces to 10 ↑ n 10 = ( 10 → 10 → n ) {\displaystyle 10\uparrow ^{n}10=(10\to 10\to n)} with an approximate n . For such numbers the advantage of using the upward arrow notation no longer applies, so

1040-713: A reasonable choice of "similar") to its current state again. Combinatorial processes give rise to astonishingly large numbers. The factorial function, which quantifies permutations of a fixed set of objects, grows exponentially as the number of objects increases. Stirling's formula provides a precise asymptotic expression for this rapid growth. In statistical mechanics, combinatorial numbers reach such immense magnitudes that they are often expressed using logarithms . Gödel numbers , along with similar representations of bit-strings in algorithmic information theory , are vast—even for mathematical statements of moderate length. Remarkably, certain pathological numbers surpass even

1105-399: A reversed version that looked exactly like a film of the first process played backwards would be equally compatible with the same fundamental laws, and would even be equally probable if one were to pick the system's initial state randomly from the phase space of all possible states for that system. Although most of the arrows of time described by physicists are thought to be special cases of

1170-426: A vertical asymptote is not helpful in defining a very large number, although the function increases very rapidly: one has to define an argument very close to the asymptote, i.e. use a very small number, and constructing that is equivalent to constructing a very large number, e.g. the reciprocal. The following illustrates the effect of a base different from 10, base 100. It also illustrates representations of numbers and

1235-497: Is able to reverse time evolution in a microscopic system, in their case of nuclear spins, which is indeed, if only for a short time, experimentally possible. In 1874, two years before the Loschmidt paper, William Thomson defended the second law against the time reversal objection in his paper "The kinetic theory of the dissipation of energy". Any process that happens regularly in the forward direction of time but rarely or never in

1300-519: Is another allowed state of motion of the system at t 1 , found by reversing all the velocities, in which H must increase. This revealed that one of Boltzmann's key assumptions, molecular chaos , or, the Stosszahlansatz , that all particle velocities were completely uncorrelated, did not follow from Newtonian dynamics. One can assert that possible correlations are uninteresting, and therefore decide to ignore them; but if one does so, one has changed

1365-434: Is applicable for transient systems, which may initially be in equilibrium and then driven away (as was the case for the first experiment by Sevick et al.) or some other arbitrary initial state, including relaxation towards equilibrium. There is also an asymptotic result for systems which are in a nonequilibrium steady state at all times. There is a crucial point in the fluctuation theorem, that differs from how Loschmidt framed

SECTION 20

#1732884403268

1430-514: Is different from Wikidata All article disambiguation pages All disambiguation pages Large numbers Scientific notation was devised to manage the vast range of values encountered in scientific research. For instance, when we write 1.0 × 10 , we express one billion —a 1 followed by nine zeros: 1,000,000,000. Conversely, the reciprocal , 1.0 × 10 , signifies one billionth, equivalent to 0.000 000 001. By using 10 instead of explicitly writing out all those zeros, readers are spared

1495-413: Is equivalent to renaming particles as antiparticles and vice versa . Therefore, this cannot explain Loschmidt's paradox. Current research in dynamical systems offers one possible mechanism for obtaining irreversibility from reversible systems. The central argument is based on the claim that the correct way to study the dynamics of macroscopic systems is to study the transfer operator corresponding to

1560-645: Is given exactly and n is an integer which may or may not be given exactly (for example: f 2 ( 3 × 10 5 ) {\displaystyle f^{2}(3\times 10^{5})} ). If n is large, any of the above can be used for expressing it. The "roundest" of these numbers are those of the form f (1) = (10→10→ m →2). For example, ( 10 → 10 → 3 → 2 ) = 10 ↑ 10 ↑ 10 10 10 10 {\displaystyle (10\to 10\to 3\to 2)=10\uparrow ^{10\uparrow ^{10^{10}}10}10} Compare

1625-501: Is large any of the above can be used to express it. Thus is obtained a nesting of forms f k m k {\displaystyle {f_{k}}^{m_{k}}} where going inward the k decreases, and with as inner argument a sequence of powers ( 10 ↑ n ) p n {\displaystyle (10\uparrow ^{n})^{p_{n}}} with decreasing values of n (where all these numbers are exactly given integers) with at

1690-568: Is large, the various representations for large numbers can be applied to this exponent itself. If this exponent is not exactly given then, again, giving a value at the right does not make sense, and instead of using the power notation of ( 10 ↑ ↑ ) {\displaystyle (10\uparrow \uparrow )} it is possible use the triple arrow operator, e.g. 10 ↑ ↑ ↑ ( 7.3 × 10 6 ) {\displaystyle 10\uparrow \uparrow \uparrow (7.3\times 10^{6})} . If

1755-410: Is obtained the somewhat counterintuitive result that a number x can be so large that, in a way, x and 10 are "almost equal" (for arithmetic of large numbers see also below). If the superscript of the upward arrow is large, the various representations for large numbers can be applied to this superscript itself. If this superscript is not exactly given then there is no point in raising the operator to

1820-499: Is possible to add 1 {\displaystyle 1} to the exponent of ( 10 ↑ ↑ ) {\displaystyle (10\uparrow \uparrow )} , to obtain e.g. ( 10 ↑ ↑ ) 3 ( 2.8 × 10 12 ) {\displaystyle (10\uparrow \uparrow )^{3}(2.8\times 10^{12})} . If the exponent of ( 10 ↑ ↑ ) {\displaystyle (10\uparrow \uparrow )}

1885-519: Is possible to have a power of the triple arrow operator. Then it is possible to proceed with operators with higher numbers of arrows, written ↑ n {\displaystyle \uparrow ^{n}} . Compare this notation with the hyper operator and the Conway chained arrow notation : An advantage of the first is that when considered as function of b , there is a natural notation for powers of this function (just like when writing out

1950-516: Is that the role of measurement is obvious in Maxwell’s demon, but not in Loschmidt’s paradox, which may explain the 40-year gap between both explanations. In the case of the single-trajectory paradox, this argument preempts the need for any other explanation, although some of them make valid points. The broader paradox, “an irreversible process cannot be deduced from reversible dynamics,” is not covered by

2015-509: Is too large to be given exactly, the value of k n + 1 {\displaystyle {k_{n+1}}} is increased by 1 and everything to the right of ( n + 1 ) k n + 1 {\displaystyle ({n+1})^{k_{n+1}}} is rewritten. For describing numbers approximately, deviations from the decreasing order of values of n are not needed. For example, 10 ↑ ( 10 ↑ ↑ ) 5

Big numbers - Misplaced Pages Continue

2080-512: Is too large to give exactly, it is possible to use a fixed n , e.g. n = 1, and apply the above recursively to m , i.e., the number of levels of upward arrows is itself represented in the superscripted upward-arrow notation, etc. Using the functional power notation of f this gives multiple levels of f . Introducing a function g ( n ) = f n ( 1 ) {\displaystyle g(n)=f^{n}(1)} these levels become functional powers of g , allowing us to write

2145-566: Is very similar to going from the latter to the sequence f n ( 10 ) {\displaystyle {f_{n}}(10)} =(10→10→10→ n ): it is the general process of adding an element 10 to the chain in the chain notation; this process can be repeated again (see also the previous section). Numbering the subsequent versions of this function a number can be described using functions f q k m q k {\displaystyle {f_{qk}}^{m_{qk}}} , nested in lexicographical order with q

2210-409: The n arrows): ( a ↑ n ) k b {\displaystyle (a\uparrow ^{n})^{k}b} . For example: and only in special cases the long nested chain notation is reduced; for ″ b ″ = 1 {\displaystyle ''b''=1} obtains: Since the b can also be very large, in general it can be written instead

2275-415: The time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behaviour of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict, hence the paradox . Josef Loschmidt's criticism

2340-569: The Gödel numbers associated with typical mathematical propositions. Logician Harvey Friedman has made significant contributions to the study of very large numbers, including work related to Kruskal's tree theorem and the Robertson–Seymour theorem . To help viewers of Cosmos distinguish between "millions" and "billions", astronomer Carl Sagan stressed the "b". Sagan never did, however, say " billions and billions ". The public's association of

2405-502: The argument given in this section. Another way of dealing with Loschmidt's paradox is to see the second law as an expression of a set of boundary conditions, in which our universe's time coordinate has a low-entropy starting point: the Big Bang . From this point of view, the arrow of time is determined entirely by the direction that leads away from the Big Bang, and a hypothetical universe with

2470-574: The arithmetic. 100 12 = 10 24 {\displaystyle 100^{12}=10^{24}} , with base 10 the exponent is doubled. 100 100 12 = 10 2 ∗ 10 24 {\displaystyle 100^{100^{12}}=10^{2*10^{24}}} , ditto. 100 100 100 12 ≈ 10 10 2 ∗ 10 24 + 0.30103 {\displaystyle 100^{100^{100^{12}}}\approx 10^{10^{2*10^{24}+0.30103}}} ,

2535-403: The chain notation can be used instead. The above can be applied recursively for this n , so the notation ↑ n {\displaystyle \uparrow ^{n}} is obtained in the superscript of the first arrow, etc., or a nested chain notation, e.g.: If the number of levels gets too large to be convenient, a notation is used where this number of levels is written down as

2600-459: The conceptual system, injecting an element of time-asymmetry by that very action. Reversible laws of motion cannot explain why we experience our world to be in such a comparatively low state of entropy at the moment (compared to the equilibrium entropy of universal heat death ); and to have been at even lower entropy in the past. Later authors have coined the term "Loschmitz's demon" (in analogy to Maxwell's demon , see below ) for an entity that

2665-689: The definition of Graham's number: it uses numbers 3 instead of 10 and has 64 arrow levels and the number 4 at the top; thus G < 3 → 3 → 65 → 2 < ( 10 → 10 → 65 → 2 ) = f 65 ( 1 ) {\displaystyle G<3\rightarrow 3\rightarrow 65\rightarrow 2<(10\to 10\to 65\to 2)=f^{65}(1)} , but also G < f 64 ( 4 ) < f 65 ( 1 ) {\displaystyle G<f^{64}(4)<f^{65}(1)} . If m in f m ( n ) {\displaystyle f^{m}(n)}

Big numbers - Misplaced Pages Continue

2730-471: The effort and potential confusion of counting an extended series of zeros to grasp the magnitude of the number. Additionally, alongside scientific notation based on powers of 10, there exists systematic nomenclature for large numbers in the short scale. Examples of large numbers describing everyday real-world objects include: In the vast expanse of astronomy and cosmology , we encounter staggering numbers related to length and time. For instance, according to

2795-510: The end a number in ordinary scientific notation. When k is too large to be given exactly, the number concerned can be expressed as f n ( 10 ) {\displaystyle {f_{n}}(10)} =(10→10→10→ n ) with an approximate n . Note that the process of going from the sequence 10 n {\displaystyle 10^{n}} =(10→ n ) to the sequence 10 ↑ n 10 {\displaystyle 10\uparrow ^{n}10} =(10→10→ n )

2860-452: The exponents first, in this case 5 > 4, so 2×10 > 5×10 . If the exponents are equal, the mantissa (or coefficient) should be compared, thus 5×10 > 2×10 because 5 > 2. Tetration with base 10 gives the sequence 10 ↑ ↑ n = 10 → n → 2 = ( 10 ↑ ) n 1 {\displaystyle 10\uparrow \uparrow n=10\to n\to 2=(10\uparrow )^{n}1} ,

2925-513: The first case, they require an increase of entropy in the measuring device that will at least offset the decrease during the reversed evolution of the gas. In the second case, Landauer's principle can be evoked to reach the same conclusion. Hence, the gas+measuring device system obeys the Second Law of Thermodynamics. It is not a coincidence that this argument mirrors closely another one given by Bennett to explain away Maxwell’s demon . The difference

2990-406: The fluctuation theorem to correctly solve the paradox. A more recent proposal concentrates on the step of the paradox in which velocities are reversed. At that moment the gas becomes an open system, and in order to reverse the velocities, position and velocity measurements have to be made. Without this, no reversal is possible. These measurements are themselves either irreversible, or reversible. In

3055-447: The highest exponent is very little more than doubled (increased by log 10 2). Loschmidt%27s paradox In physics , Loschmidt's paradox (named for J.J. Loschmidt ), also known as the reversibility paradox , irreversibility paradox , or Umkehreinwand (from German  'reversal objection'), is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts

3120-417: The lower-tower representation of the log 10 of the whole number. If the power tower would contain one or more numbers different from 10, the two approaches would lead to different results, corresponding to the fact that extending the power tower with a 10 at the bottom is then not the same as extending it with a 10 at the top (but, of course, similar remarks apply if the whole power tower consists of copies of

3185-405: The microscopic equations of motion. It is then argued that the transfer operator is not unitary ( i.e. is not reversible) but has eigenvalues whose magnitude is strictly less than one; these eigenvalues corresponding to decaying physical states. This approach is fraught with various difficulties; it works well for only a handful of exactly solvable models. Abstract mathematical tools used in

3250-401: The most significant number, but with decreasing order for q and for k ; as inner argument yields a sequence of powers ( 10 ↑ n ) p n {\displaystyle (10\uparrow ^{n})^{p_{n}}} with decreasing values of n (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation. For

3315-460: The observable universe. According to Don Page , physicist at the University of Alberta, Canada, the longest finite time that has so far been explicitly calculated by any physicist is which corresponds to the scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming

SECTION 50

#1732884403268

3380-513: The opposite direction, such as entropy increasing in an isolated system, defines what physicists call an arrow of time in nature. This term only refers to an observation of an asymmetry in time; it is not meant to suggest an explanation for such asymmetries. Loschmidt's paradox is equivalent to the question of how it is possible that there could be a thermodynamic arrow of time given time-symmetric fundamental laws, since time-symmetry implies that for any process compatible with these fundamental laws,

3445-431: The paradox. Loschmidt considered the probability of observing a single trajectory, which is analogous to enquiring about the probability of observing a single point in phase space. In both of these cases the probability is always zero. To be able to effectively address this you must consider the probability density for a set of points in a small region of phase space, or a set of trajectories. The fluctuation theorem considers

3510-533: The phrase and Sagan came from a Tonight Show skit. Parodying Sagan's effect, Johnny Carson quipped "billions and billions". The phrase has, however, now become a humorous fictitious number—the Sagan . Cf. , Sagan Unit . A standardized way of writing very large numbers allows them to be easily sorted in increasing order, and one can get a good idea of how much larger a number is than another one. To compare numbers in scientific notation, say 5×10 and 2×10 , compare

3575-448: The power towers of numbers 10, where ( 10 ↑ ) n {\displaystyle (10\uparrow )^{n}} denotes a functional power of the function f ( n ) = 10 n {\displaystyle f(n)=10^{n}} (the function also expressed by the suffix "-plex" as in googolplex, see the googol family ). These are very round numbers, each representing an order of magnitude in

3640-474: The prevailing Big Bang model , our universe is approximately 13.8 billion years old (equivalent to 4.355 × 10^17 seconds). The observable universe spans an incredible 93 billion light years (approximately 8.8 × 10^26 meters) and hosts around 5 × 10^22 stars, organized into roughly 125 billion galaxies (as observed by the Hubble Space Telescope). As a rough estimate, there are about 10^80 atoms within

3705-409: The probability density for all of the trajectories that are initially in an infinitesimally small region of phase space. This leads directly to the probability of finding a trajectory, in either the forward or the reverse trajectory sets, depending upon the initial probability distribution as well as the dissipation which is done as the system evolves. It is this crucial difference in approach that allows

3770-825: The right-hand argument of the triple arrow operator is large the above applies to it, obtaining e.g. 10 ↑ ↑ ↑ ( 10 ↑ ↑ ) 2 ( 10 ↑ ) 497 ( 9.73 × 10 32 ) {\displaystyle 10\uparrow \uparrow \uparrow (10\uparrow \uparrow )^{2}(10\uparrow )^{497}(9.73\times 10^{32})} (between 10 ↑ ↑ ↑ 10 ↑ ↑ ↑ 4 {\displaystyle 10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 4} and 10 ↑ ↑ ↑ 10 ↑ ↑ ↑ 5 {\displaystyle 10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 5} ). This can be done recursively, so it

3835-458: The same number, different from 10). If the height of the tower is large, the various representations for large numbers can be applied to the height itself. If the height is given only approximately, giving a value at the top does not make sense, so the double-arrow notation (e.g. 10 ↑ ↑ ( 7.21 × 10 8 ) {\displaystyle 10\uparrow \uparrow (7.21\times 10^{8})} ) can be used. If

3900-419: The same term [REDACTED] This disambiguation page lists articles associated with the title Big numbers . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Big_numbers&oldid=1062122087 " Category : Disambiguation pages Hidden categories: Short description

3965-436: The study of dissipative systems include definitions of mixing , wandering sets , and ergodic theory in general. One approach to handling Loschmidt's paradox is the fluctuation theorem , derived heuristically by Denis Evans and Debra Searles , which gives a numerical estimate of the probability that a system away from equilibrium will have a certain value for the dissipation function (often an entropy like property) over

SECTION 60

#1732884403268

4030-447: The thermodynamic arrow, there are a few that are believed to be unconnected, like the cosmological arrow of time based on the fact that the universe is expanding rather than contracting, and the fact that a few processes in particle physics actually violate time-symmetry, while they respect a related symmetry known as CPT symmetry . In the case of the cosmological arrow, most physicists believe that entropy would continue to increase even if

4095-400: The universe began to contract (although the physicist Thomas Gold once proposed a model in which the thermodynamic arrow would reverse in this phase). In the case of the violations of time-symmetry in particle physics, the situations in which they occur are rare and are only known to involve a few types of meson particles. Furthermore, due to CPT symmetry , reversal of the direction of time

4160-458: The value after the double arrow is a very large number itself, the above can recursively be applied to that value. Examples: Similarly to the above, if the exponent of ( 10 ↑ ) {\displaystyle (10\uparrow )} is not exactly given then giving a value at the right does not make sense, and instead of using the power notation of ( 10 ↑ ) {\displaystyle (10\uparrow )} , it

4225-472: Was provoked by the H-theorem of Boltzmann , which employed kinetic theory to explain the increase of entropy in an ideal gas from a non-equilibrium state, when the molecules of the gas are allowed to collide. In 1876, Loschmidt pointed out that if there is a motion of a system from time t 0 to time t 1 to time t 2 that leads to a steady decrease of H (increase of entropy ) with time, then there

#267732