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Statistical physics books often motivate the necessity of statistical/thermodynamic description by impossibility of calculating the trajectories of all the molecules (I speak of "trajectories" but the discussion applies to the quantum case with the obvious adjustments). The bases of the statistical physics were laid about a hundred years ago, when the limited computation abilities seemed indeed to be an unsurmountable obstacle. The situation has however changed since then and it is not uncommon to suggest that we could in principle attain enough computational power to predict behavior of any system of interest (with the corollary that such ab initio calculation would render redundant the empirical laws of chemistry, biology, economics, psychology, as was debated in the comments that followed the question on whether physics explains biology.)

The traditional answer to this question is to emphasize that the systems
considered in thermodynamics are so complex (they contain a large number of
interacting particles) that we are obliged to introduce approximations. The
Second Law of thermodynamics would have its roots in these approximations!
Some authors go so far as to state that entropy is only the expression of our
ignorance!
Here again the recent extension of thermodynamics to situations far from
equilibrium is essential. Irreversible processes lead then to new space-time
structures. They play therefore a basic constructive role. No life would be
possible without irreversible processes (Chapter 19). It seems absurd to suggest
that life would be the result of our approximations! We therefore cannot deny the
reality of entropy, the very essence of an arrow of time in nature. We are the
children of evolution and not its progenitors.

Statistical physics books often motivate the necessity of statistical/thermodynamic description by impossibility of calculating the trajectories of all the molecules (I speak of "trajectories" but the discussion applies to the quantum case with the obvious adjustments). The bases of the statistical physics were laid about a hundred years ago, when the limited computation abilities seemed indeed to be an insurmountable obstacle. The situation has however changed since then and it is not uncommon to suggest that we could in principle attain enough computational power to predict behavior of any system of interest (with the corollary that such ab initio calculation would render redundant the empirical laws of chemistry, biology, economics, psychology, as was debated in the comments that followed the question on whether physics explains biology.)

The traditional answer to this question is to emphasize that the systems considered in thermodynamics are so complex (they contain a large number of interacting particles) that we are obliged to introduce approximations. The Second Law of thermodynamics would have its roots in these approximations! Some authors go so far as to state that entropy is only the expression of our ignorance! Here again the recent extension of thermodynamics to situations far from equilibrium is essential. Irreversible processes lead then to new space-time structures. They play therefore a basic constructive role. No life would be possible without irreversible processes (Chapter 19). It seems absurd to suggest that life would be the result of our approximations! We therefore cannot deny the reality of entropy, the very essence of an arrow of time in nature. We are the children of evolution and not its progenitors.

Update 5
In a metaphoric form the problem that we face here was famously described by Jorge Luis Borges in their short story On the Exactitude of Science (the full text is taken from here):

...In that Empire, the Art of Cartography attained such Perfection that the map of a single Province occupied the entirety of a City, and the map of the Empire, the entirety of a Province. In time, those Unconscionable Maps no longer satisfied, and the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it. The following Generations, who were not so fond of the Study of Cartography as their Forebears had been, saw that that vast Map was Useless, and not without some Pitilessness was it, that they delivered it up to the Inclemencies of Sun and Winters. In the Deserts of the West, still today, there are Tattered Ruins of that Map, inhabited by Animals and Beggars; in all the Land there is no other Relic of the Disciplines of Geography.
—Suarez Miranda,Viajes de varones prudentes, Libro IV,Cap. XLV, Lerida, 1658

Update 4
Interesting relevant quote from Kondepudi & Prigogine's book Modern Thermodynamics:

The traditional answer to this question is to emphasize that the systems
considered in thermodynamics are so complex (they contain a large number of
interacting particles) that we are obliged to introduce approximations. The
Second Law of thermodynamics would have its roots in these approximations!
Some authors go so far as to state that entropy is only the expression of our
ignorance!
Here again the recent extension of thermodynamics to situations far from
equilibrium is essential. Irreversible processes lead then to new space-time
structures. They play therefore a basic constructive role. No life would be
possible without irreversible processes (Chapter 19). It seems absurd to suggest
that life would be the result of our approximations! We therefore cannot deny the
reality of entropy, the very essence of an arrow of time in nature. We are the
children of evolution and not its progenitors.

Statistical physics books often motivate the necessity of statistical/thermodynamic description by impossibility of calculating the trajectories of all the molecules (I speak of "trajectories" but the discussion applies to the quantum case with the obvious adjustments). The bases of the statistical physics were laid about a hundred years ago, when the limited computation abilities seemed indeed to be an unsurmountable obstacle. The situation has however changed since then and it is not uncommon to suggest that we could in principle attain enough computational power to predict behavior of any system of interest (with the corollary that such ab initio calculation would render redundant the empirical laws of chemistry, biology, economics, psychology, as was debated in the comments that followed the question on whether physics explains biology.)

One could oppose this view by noting that, at a certain scale, the computer itself, being the part of the Universe, would interfere with such a calculation. This might be compounded by the possible non-linearities, which can result in small fluctuations producing wildly different results. Moreover, we are not free to choose the initial state of our computation, given the trajectory taken by the Universe from the Big Bang till now.

In a sense this limitation is similar to those that gave rise to relativity and quantum mechanics: the observer (the computer in this case) influences the outcome of measurement (computation).

The question, to which I would like to obtain a factual answer is:
Is the above described computational capacity limitation fundamental, or not, or it is impossible to say?

Update
A related question: more specifically focused on molecular physics rather than the general problem of computability.

Update 2
Just in case the comments are removed for some reason, I would like to put here the valuable references given by @ChiralAnomaly:

• S. Aaronson, NP-complete problems and physicsl reality
• S. Aaronson, Limits on efficient computation in physical world
• L. Susskind, Computational complexity and black hole horizons

Statistical physics books often motivate the necessity of statistical/thermodynamic description by impossibility of calculating the trajectories of all the molecules (I speak of "trajectories" but the discussion applies to the quantum case with the obvious adjustments). The bases of the statistical physics were laid about a hundred years ago, when the limited computation abilities seemed indeed to be an unsurmountable obstacle. The situation has however changed since then and it is not uncommon to suggest that we could in principle attain enough computational power to predict behavior of any system of interest (with the corollary that such ab initio calculation would render redundant the empirical laws of chemistry, biology, economics, psychology, as was debated in the comments that followed the question on whether physics explains biology.)

One could oppose this view by noting that, at a certain scale, the computer itself, being the part of the Universe, would interfere with such a calculation. This might be compounded by the possible non-linearities, which can result in small fluctuations producing wildly different results. Moreover, we are not free to choose the initial state of our computation, given the trajectory taken by the Universe from the Big Bang till now.

In a sense this limitation is similar to those that gave rise to relativity and quantum mechanics: the observer (the computer in this case) influences the outcome of measurement (computation).

The question, to which I would like to obtain a factual answer is:
Is the above described computational capacity limitation fundamental, or not, or it is impossible to say?

Update
A related question: more specifically focused on molecular physics rather than the general problem of computability.

Update 2
Just in case the comments are removed for some reason, I would like to put here the valuable references given by @ChiralAnomaly:

• S. Aaronson, NP-complete problems and physicsl reality
• S. Aaronson, Limits on efficient computation in physical world
• L. Susskind, Computational complexity and black hole horizons

Update 3
To restate the question differently:
Can we model complex (e.g., biological or social) systems without resorting to intermediate concepts ("emergent properties"), directly in terms of basic physical equations.
(This formulation suggests an answer along the lines of the information loss when replacing detailed description by a higher level concept, and the inevitable noise resulting from such an approximate description, which makes explanations in terms of basic concepts meaningless.)