Talk:Fermi-Pasta-Ulam nonlinear lattice oscillations

This contribution needs significant rewriting, according to my understanding. Below are the points listed in the order of their appearance in the text.

1. Caption Fig.1

Replace '/N' by '/(N+1)' in all denominators (three times).

2. Section 'In real space: ...'

Equation (2) is derived for periodic boundary conditions (PBC) as opposed to fixed boundary conditions (FBC) in the FPU studies. Also it is just one of two (first order in time) equations, which are derived for right and left going waves. Both equations have to be combined in order to obtain the original field u_n.

Section 'In normal mode space ...'

First I think that this section should contain a much more detailed discussion of the various results on dynamics in normal mode space, which is right now squeezed into the section 'Recent approaches'. These studies clearly show (and not just 'propose' that something 'could') that the FPU trajectory DOES relax to equipartition. The issue is about at least two time scales - on a short one tau_1 a (typically exponentially) localized packet of energy distributions in mode space is reached (which is remarkably close to the profile when finding a periodic orbit, see Chaos 17, 023102 (2007) ). On a second, much larger scale tau_2 the system finally reaches equipartition. This scale tau_2 seems to depend sensitively on control parameters, and can be brought down to tau_1. Thus the FPU problem from a present perspective is: i) why are there two time scales in the relaxation? ii) what is the profile of the localized packet in mode space? iii) how does tau_2 depend on the control parameters?

Figure 3 is for alpha FPU, not beta FPU, as written in the caption. Also it is taken from a recent publication by Zabusky et al. This should be corrected respectively cited. Also the initial condition in Fig.4 is NOT a normal mode, but a wave packet, which is quite localized in real space. Therefore it is not a pi mode. Also note, that a pi mode IS an exact normal mode for PBC, but is NOT for FBC. Apparently in Fig.3 we see a simulation for PBC. This should be appropriately mentioned.

4. Section 'Relaxation to equipartition for short ...'

This section is misleading. When repeating the FPU type experiment, for weak enough nonlinearities, one again observes a localized distribution in normal mode space up to some long time scale tau_2 (e.g. Chaos 17, 023102 (2007) and references therein). Therefore there is no conceptual change here. However, when reaching the time scale tau_2, instead of equipartition, breathers in real space may form. I would suggest to clarify these issues. Breathers, or thermal equilibrium right after tau_2, is reached depending on short or long wavelength excitations. But the FPU problem (what happens BEFORE tau_2) is the same.

5. Section 'Recent approaches'

As already suggested above, most of these rather short and vague statements can be substantiated, and discussed in the above sections on normal mode space. This concerns also the results on periodic orbits, see http://www.pks.mpg.de/~flach/html/preprints_q_breathers.html , and in particular Physica D 237, 908 (2008), Am. J. Phys. 76, 453 (2008) and Int. J. Mod. Phys. B 21, 3925 (2007), where comparisons with resonant normal form analysis, and scaling properties of periodic orbits, are discussed.

======= Reviewer C ===========

This is a very nice contribution, clear and well written. I agree with the comments of the other reviewers and with the suggested changes. In particular, it is of fundamental importance that the FPU model DOES relax to equipartition, as reviewer A has remarked. However, at finite N it has been shown in: L. Casetti, M. Cerruti-Sola, M. Pettini and E.G.D. Cohen, "The Fermi-Pasta-Ulam problem revisited: Stochasticity thresholds in nonlinear Hamiltonian systems",Phys. Rev.E55, 6566 (1997), that a very steep drop of the largest Lyapunov exponent suggests the existence of a stochasticity threshold (in principle a stochastic layer is always there, but its measure seems to be suddenly shrunk to a very small value). This stochasticity threshold vanishes, in energy density, as 1/N^2.

In the section "In normal mode space: chaotic properties and Chirikov's resonance overlap criterion", the authors mention the so-called strong stochasticity threshold, but the quotation: Livi R. et al. (1985) "Equipartition threshold in nonlinear large Hamiltonian systems: The Fermi-Pasta-Ulam model" Phys. Rev. A 31:1039-1045, is not appropriate there. The definition "strong stochasticity threshold" has been introduced for the first time in the papers: M. Pettini and M. Landolfi, "Relaxation properties and ergodicity breaking in nonlinear hamiltonian dynamics", Phys. Rev. A41, 768 (1990), and M. Pettini and M. Cerruti-Sola, "Strong stochasticity threshold in nonlinear large Hamiltonian systems: Effect on mixing times", Phys. Rev. A44, 975 (1991), where a crossover in the energy density scaling of the largest Lyapunov exponent has been fist put in evidence, together with the fact that, in correspondence with this crossover, the relaxation time to equipartition of a nonequilibrium initial condition also sharply changes its pattern (the relaxation times are steeply increasing by lowering the energy density below the crossover of the largest LE).

I would ask the authors to emend the referencing according to this remark.

======= Reviewer D ===========

The article is written in a very good way, however, there are points to be corrected. Apart from the technical misprints in the formulas and figure captions indicated by another Referee, I would add the following suggestions:

1. I think it is better to interchange the figures 1 and 2, according to

  their appearance in the text.

2. I suggest to indicate that the statistical behavior (relaxation

 to the steady-state distribution) was expected since the number
 of particle was quite large. Otherwise, for small number of particles
one should not expect that the statistical mechanics is valid. 
 For example, instead of 

 "... contrary to the predictions of statistical mechanics, the energy..."

 it is better to write,

 "...contrary to the predictions of statistical mechanics for $N$ \goto \infty, ..."

 or, something like that.

3. The sentence:

 "In 1965, using the so-called continuum limit, Zabusky and Kruskal 
 (Zabusky and Kruskal 1965) were able to explain the periodic ..."
 seems to me a bit strong. I would say that more correct way is to say
 "... were able to RELATE the periodic ...". I mean that the continuous
 model where the soliton solution was found (first, numerically, and 
 after, analytically) is quite far from that in the discrete FPU model.  
 It is important that in the KdV model the interaction 
 between the waves traveling in opposite directions is absent, in 
 contrast with the original FPU model. It was shown numerically that 
 this interaction is crucial and directly related to the onset 
 of chaos (see details in the paper Comp. Phys. Comm. 5 (1973) 11-16). 
 In my opinion, one needs to indicate this point.

4. When discussing the short wavelength limit, the impression is that the dynamics

  in this case can be explained only in terms of breathers. However, the
 approach based on the overlap of nonlinear resonances gives a correct
 prediction for the chaos border in this case as well. In particular, it seems to me
 not good to say that "... the pathway to equipartition LEADS to the 
 creation of highly localized excitations..." since this depends on the
 initial conditions. I suggest to write more carefully "... the pathway 
 to equipartition MAY LEAD to the creation of highly localized excitations..".

Author Ruffo :

We thank the referees for their careful reading of the text.

As explained below, we agree with many of the comments. However, some comments refer to too specific points from our point of view, and we are convinced that we should not go so deeply in a general article.

Thierry Dauxois and Stefano Ruffo

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Reply to Reviewer A

   This contribution needs significant rewriting, according to my
   understanding. Below are the points listed in the order of their
   appearance in the text.
   1. Caption Fig.1: Replace '/N' by '/(N+1)' in all denominators.

Yes. We modified the text accordingly.

   2. Section 'In real space: ...'
   Equation (2) is derived for periodic boundary conditions (PBC) as
   opposed to fixed boundary conditions (FBC) in the FPU studies.

It is correct and we have amended the text accordingly.

   Also it is just one of two (first order in time) equations, which are
   derived for right and left going waves. Both equations have to be
   combined in order to obtain the original field u_n. Therefore there
   is a more complicated transformation from w to u_n, it seems.

We agree (see also the reply to Referee D). We have modified the text. However, the derivation of the two KdV uncoupled equations for right-going and left-going waves is for us too detailed and not necessary for a Scholarpedia article. We have added a Reference.

    The legend in Fig.3 contains u, which can be easily misinterpreted
    as the original field u_n, while it should be w. I suggest to change
    the notations in the text, or to explain this point clearly in the caption.

We were already using w and not u in the legend in Fig. 3.

   It is interesting to note that exciting the k=1 mode in the FPU
   chain for weak nonlinearity alpha will NOT lead to the soliton train
   observed for a single KdV equation, but to a much smoother field
   wiggling in real space.

It's certainly correct, but we prefer to put the historical simulation by Zabusky on the PBC problem giving the KdV like behavior.

   It is the normal mode space (k) where the clear notion of an
   exponentially localized distribution of energies is observed.

We quote the exponential localization in the last section on "Recent Approaches" and it is definitely an interesting result. Again, we do not want to be too detailed here.

   Figure 3 is for alpha FPU, not beta FPU, as written in the caption.

Maybe, the Reviewer was meaning Fig. 4, where we were indeed wrong.

   Also it is taken from a recent publication by Zabusky et al. This
   should be corrected respectively cited.
   Also the initial condition in Fig.4 is NOT a normal mode, but a wave
   packet, which is quite localized in real space. Therefore it is not
   a pi mode. Also note, that a pi mode IS an exact normal mode for
   PBC, but is NOT for FBC.

We agree. Rather than describing separately Figs. 4 and 5, we have decided, for reasons of clarity, to erase Fig. 4, maintaining only one Figure for the description of the pi-mode evolution with PBC.

   Apparently in Fig.3 we see a simulation for PBC. This should be
   appropriately mentioned.

It was already mentioned in the previous version.

   Reading the review paper by Ford more carefully would show, that the
   FPU problem remained a mystery (at least to Ford) up to 1992. For
   good reasons, as explained there.

We have quoted Ford's review paper. This remark is also very mysterious for us!

   3. Section 'In normal mode space ...'
   First I think that this section should contain a much more detailed
   discussion of the various results on dynamics in normal mode space,
   which is right now squeezed into the section 'Recent approaches'.

This is not true, because the first section starts by discussing the FPU paradox in normal modes. It's true that there has been important recent work in normal mode space (the "two time scales theory", q-breathers, etc.) but it would not be appropriate to discuss this issues in the context of an historical account of the FPU problem. Maybe, this could be the matter for another Scholarpedia entry.

   These studies clearly show (and not just 'propose' that something
   'could') that the FPU trajectory DOES relax to equipartition. 

As Reviewer C mentions, this issue is not at all clarified, and it could well be that the FPU initial condition was chose in an ordered region of phase space, preventing the relaxation to equipartition. This this question is still debated, we prefer not to make strong statements.

   The issue is about at least two time scales - on a short one tau_1 a
   (typically exponentially) localized packet of energy distributions
   in mode space is reached (which is remarkably close to the profile
   when finding a periodic orbit, see Chaos 17, 023102 (2007) ). On a
   second, much larger scale tau_2 the system finally reaches
   equipartition. This scale tau_2 seems to depend sensitively on
   control parameters, and can be brought down to tau_1. Thus the FPU
   problem from a present perspective is: i) why are there two time
   scales in the relaxation? ii) what is the profile of the localized
   packet in mode space? iii) how does tau_2 depend on the control
   parameters?

As we have remarked above, although we agree that these are important issues (and indeed we briefly quoted the two-time-scales theory in the section on "Recent approaches"), we believe is not appropriate to expand it too much in this context.

   4. Section 'Relaxation to equipartition for short ...'
   This section is misleading. When repeating the FPU type experiment,
   for weak enough nonlinearities, one again observes a localized
   distribution in normal mode space up to some long time scale tau_2
   (e.g. Chaos 17, 023102 (2007) and references therein). Therefore
   there is no conceptual change here. However, when reaching the time
   scale tau_2, instead of equipartition, breathers in real space may
   form. I would suggest to clarify these issues. Breathers, or thermal
   equilibrium right after tau_2, is reached depending on short or long
   wavelength excitations. But the FPU problem (what happens BEFORE
   tau_2) is the same.

   5. Section 'Recent approaches'
   As already suggested above, most of
   these rather short and vague statements can be substantiated, and
   discussed in the above sections on normal mode space. This concerns
   also the results on periodic orbits, see
   http://www.pks.mpg.de/~flach/html/preprints_q_breathers.html , and
   in particular Physica D 237, 908 (2008), Am. J. Phys. 76, 453 (2008)
   and Int. J. Mod. Phys. B 21, 3925 (2007), where comparisons with
   resonant normal form analysis, and scaling properties of periodic
   orbits, are discussed.

In the "Instructions for authors" it is emphasized that "To be authoritative, the article should provide the full description of the topic, possibly ending with the details useful for experts." It is clearly what we have proposed.

It's clear that the referee wants to defend the recent and very interesting concept of q-breathers, but their relevance for the FPU relaxation problem is still debated and, from our point of view, they should be maintained in the last section we wrote. These papers are interesting but definitely belong to the 'Recent approaches' section.

We are convinced that the reader will look first for the definition of the FPU problem, and then for the two main (and old) approaches that are the solitons and the Resonance overlap criterion. Only experts will be interested in the others approached developed in the last few years. Anyway, we have added the Reference to Penati and Flach paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Reply to Reviewer C

  This is a very nice contribution, clear and well written. I agree 
  with the comments of the other reviewers and with the suggested changes.
  In particular, it is of fundamental importance that the FPU model DOES
  relax to equipartition, as reviewer A has remarked.
  However, at finite N it has been shown in: L. Casetti, M. Cerruti-Sola, 
  M. Pettini and E.G.D. Cohen, "The Fermi-Pasta-Ulam problem revisited: 
  Stochasticity thresholds in nonlinear Hamiltonian systems",Phys. Rev.E55, 
  6566 (1997), that a very steep drop of the largest Lyapunov exponent suggests 
  the existence of a stochasticity threshold (in principle a stochastic layer
  is always there, but its measure seems to be suddenly shrunk to a very small value).
  This stochasticity threshold vanishes, in energy density, as 1/N^2.
  In the section "In normal mode space: chaotic properties and Chirikov's 
  resonance overlap criterion", the authors mention the so-called
  strong stochasticity threshold, but the quotation: 
  Livi R. et al. (1985) "Equipartition threshold in nonlinear large Hamiltonian 
  systems: The Fermi-Pasta-Ulam model" Phys. Rev. A 31:1039-1045, is not
  appropriate there. The definition "strong stochasticity threshold" has been
  introduced for the first time in the papers: M. Pettini and M. Landolfi, 
  "Relaxation properties and ergodicity breaking in nonlinear hamiltonian dynamics",
  Phys. Rev. A41, 768 (1990), and M. Pettini and M. Cerruti-Sola, "Strong stochasticity 
  threshold in nonlinear large Hamiltonian systems: Effect on mixing times", 
  Phys. Rev. A44, 975 (1991),
  where a crossover in the energy density scaling of the largest Lyapunov exponent
  has been fist put in evidence, together with the fact that, in correspondence with
  this crossover, the relaxation time to equipartition of a nonequilibrium initial 
  condition also sharply changes its pattern (the relaxation times are steeply
  increasing by lowering the energy density below the crossover of the largest LE).
  I would ask the authors to emend the referencing according to this remark.

We have added a sentence and a quote to these papers.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Reply to Reviewer D

  The article is written in a very good way, however, there are points to be
  corrected. Apart from the technical misprints in the formulas and figure
  captions indicated by another Referee, I would add the following suggestions:
  1. I think it is better to interchange the figures 1 and 2, according to
  their appearance in the text.

We have modified the order of these two pictures.

  2. I suggest to indicate that the statistical behavior (relaxation
  to the steady-state distribution) was expected since the number
  of particle was quite large. Otherwise, for small number of particles
  one should not expect that the statistical mechanics is valid.
  For example, instead of
  "... contrary to the predictions of statistical mechanics, the energy..."
  it is better to write,
  "...contrary to the predictions of statistical mechanics for
  $N$ \goto \infty, ..." or, something like that.

We have introduced the modification "when the number of particles\(N\) is going to infinity"

3. The sentence:
"In 1965, using the so-called continuum limit, Zabusky and Kruskal
(Zabusky and Kruskal 1965) were able to explain the periodic ..."
seems to me a bit strong. I would say that more correct way is to say
"... were able to RELATE the periodic ...". I mean that the continuous
model where the soliton solution was found (first, numerically, and
after, analytically) is quite far from that in the discrete FPU model.
It is important that in the KdV model the interaction
between the waves traveling in opposite directions is absent, in
contrast with the original FPU model. It was shown numerically that
this interaction is crucial and directly related to the onset
of chaos (see details in the paper Comp. Phys. Comm. 5 (1973) 11-16).
In my opinion, one needs to indicate this point.

We have made the modification and introduced the reference.

4. When discussing the short wavelength limit, the impression is that the
dynamics in this case can be explained only in terms of breathers.
However, the approach based on the overlap of nonlinear resonances gives
a correct prediction for the chaos border in this case as well.
In particular, it seems to me not good to say that "... the pathway
to equipartition LEADS to the creation of highly localized excitations..."
since this depends on the initial conditions. I suggest to write more
carefully "... the pathway
to equipartition MAY LEAD to the creation of highly localized excitations..".

We have followed this suggestion. We have added a sentence, since indeed we had done ourselves this remark in the paper published in Chaos in 2005.

Reviewer A

Some of my previous comments have not been incorporated. I list them below (slightly modified, to may be give better understanding).

1. Section 'In real space: ...'

It is interesting to note that e.g. exciting the k=1 mode in the FPU chain for weak nonlinearity alpha will NOT lead to the soliton train observed for a single KdV equation, but to a much smoother field wiggeling in real space. It is the normal mode space (k) where the clear notion of an exponentially localized distribution of energies is observed. WITHOUT THIS INFORMATION, THE CHAPTER IS INCOMPLETE, AND THE READER MAY ERRONEOUSLY CONCLUDE, THAT SOLITONS IN REAL SPACE ARE THE CORRECT EXPLANATION OF THE FPU OBSERVATIONS. IN PARTICULAR, THE RECURRENCE PHENOMENON IS NOTHING BUT BEATING BETWEEN A FEW EXCITED DEGREES OF FREEDOM - WHATEVER THEIR NATURE. THEREFORE, THE LAST SENTENCE OF THAT SECTION IS MISLEADING AND PERHAPS EVEN WRONG. Now, I do not want the authors to change their taste, but I think it is fair to point exactly at the limits where the soliton picture disappears and breaks down altogether.

2. Section 'Recent approaches'

'It has been proposed that ... conditions could proceed ...' is wrong. Correct would be 'It has been observed ... conditions proceed ...'. These are reported facts, and not just suggestions.

understanding. Below are the points listed in the order of their appearance in the text.

1. Caption Fig.1

Replace '/N' by '/(N+1)' in all denominators (three times).

2. Section 'In real space: ...'

Equation (2) is derived for periodic boundary conditions (PBC) as opposed to fixed boundary conditions (FBC) in the FPU studies. Also it is just one of two (first order in time) equations, which are derived for right and left going waves. Both equations have to be combined in order to obtain the original field u_n.

Section 'In normal mode space ...'

First I think that this section should contain a much more detailed discussion of the various results on dynamics in normal mode space, which is right now squeezed into the section 'Recent approaches'. These studies clearly show (and not just 'propose' that something 'could') that the FPU trajectory DOES relax to equipartition. The issue is about at least two time scales - on a short one tau_1 a (typically exponentially) localized packet of energy distributions in mode space is reached (which is remarkably close to the profile when finding a periodic orbit, see Chaos 17, 023102 (2007) ). On a second, much larger scale tau_2 the system finally reaches equipartition. This scale tau_2 seems to depend sensitively on control parameters, and can be brought down to tau_1. Thus the FPU problem from a present perspective is: i) why are there two time scales in the relaxation? ii) what is the profile of the localized packet in mode space? iii) how does tau_2 depend on the control parameters?

Figure 3 is for alpha FPU, not beta FPU, as written in the caption. Also it is taken from a recent publication by Zabusky et al. This should be corrected respectively cited. Also the initial condition in Fig.4 is NOT a normal mode, but a wave packet, which is quite localized in real space. Therefore it is not a pi mode. Also note, that a pi mode IS an exact normal mode for PBC, but is NOT for FBC. Apparently in Fig.3 we see a simulation for PBC. This should be appropriately mentioned.

4. Section 'Relaxation to equipartition for short ...'

This section is misleading. When repeating the FPU type experiment, for weak enough nonlinearities, one again observes a localized distribution in normal mode space up to some long time scale tau_2 (e.g. Chaos 17, 023102 (2007) and references therein). Therefore there is no conceptual change here. However, when reaching the time scale tau_2, instead of equipartition, breathers in real space may form. I would suggest to clarify these issues. Breathers, or thermal equilibrium right after tau_2, is reached depending on short or long wavelength excitations. But the FPU problem (what happens BEFORE tau_2) is the same.

5. Section 'Recent approaches'

As already suggested above, most of these rather short and vague statements can be substantiated, and discussed in the above sections on normal mode space. This concerns also the results on periodic orbits, see http://www.pks.mpg.de/~flach/html/preprints_q_breathers.html , and in particular Physica D 237, 908 (2008), Am. J. Phys. 76, 453 (2008) and Int. J. Mod. Phys. B 21, 3925 (2007), where comparisons with resonant normal form analysis, and scaling properties of periodic orbits, are discussed.

======= Reviewer C ===========

This is a very nice contribution, clear and well written. I agree with the comments of the other reviewers and with the suggested changes. In particular, it is of fundamental importance that the FPU model DOES relax to equipartition, as reviewer A has remarked. However, at finite N it has been shown in: L. Casetti, M. Cerruti-Sola, M. Pettini and E.G.D. Cohen, "The Fermi-Pasta-Ulam problem revisited: Stochasticity thresholds in nonlinear Hamiltonian systems",Phys. Rev.E55, 6566 (1997), that a very steep drop of the largest Lyapunov exponent suggests the existence of a stochasticity threshold (in principle a stochastic layer is always there, but its measure seems to be suddenly shrunk to a very small value). This stochasticity threshold vanishes, in energy density, as 1/N^2.

In the section "In normal mode space: chaotic properties and Chirikov's resonance overlap criterion", the authors mention the so-called strong stochasticity threshold, but the quotation: Livi R. et al. (1985) "Equipartition threshold in nonlinear large Hamiltonian systems: The Fermi-Pasta-Ulam model" Phys. Rev. A 31:1039-1045, is not appropriate there. The definition "strong stochasticity threshold" has been introduced for the first time in the papers: M. Pettini and M. Landolfi, "Relaxation properties and ergodicity breaking in nonlinear hamiltonian dynamics", Phys. Rev. A41, 768 (1990), and M. Pettini and M. Cerruti-Sola, "Strong stochasticity threshold in nonlinear large Hamiltonian systems: Effect on mixing times", Phys. Rev. A44, 975 (1991), where a crossover in the energy density scaling of the largest Lyapunov exponent has been fist put in evidence, together with the fact that, in correspondence with this crossover, the relaxation time to equipartition of a nonequilibrium initial condition also sharply changes its pattern (the relaxation times are steeply increasing by lowering the energy density below the crossover of the largest LE).

I would ask the authors to emend the referencing according to this remark.

======= Reviewer D ===========

The article is written in a very good way, however, there are points to be corrected. Apart from the technical misprints in the formulas and figure captions indicated by another Referee, I would add the following suggestions:

1. I think it is better to interchange the figures 1 and 2, according to

  their appearance in the text.

2. I suggest to indicate that the statistical behavior (relaxation

 to the steady-state distribution) was expected since the number
 of particle was quite large. Otherwise, for small number of particles
one should not expect that the statistical mechanics is valid. 
 For example, instead of 

 "... contrary to the predictions of statistical mechanics, the energy..."

 it is better to write,

 "...contrary to the predictions of statistical mechanics for $N$ \goto \infty, ..."

 or, something like that.

3. The sentence:

 "In 1965, using the so-called continuum limit, Zabusky and Kruskal 
 (Zabusky and Kruskal 1965) were able to explain the periodic ..."
 seems to me a bit strong. I would say that more correct way is to say
 "... were able to RELATE the periodic ...". I mean that the continuous
 model where the soliton solution was found (first, numerically, and 
 after, analytically) is quite far from that in the discrete FPU model.  
 It is important that in the KdV model the interaction 
 between the waves traveling in opposite directions is absent, in 
 contrast with the original FPU model. It was shown numerically that 
 this interaction is crucial and directly related to the onset 
 of chaos (see details in the paper Comp. Phys. Comm. 5 (1973) 11-16). 
 In my opinion, one needs to indicate this point.

4. When discussing the short wavelength limit, the impression is that the dynamics

  in this case can be explained only in terms of breathers. However, the
 approach based on the overlap of nonlinear resonances gives a correct
 prediction for the chaos border in this case as well. In particular, it seems to me
 not good to say that "... the pathway to equipartition LEADS to the 
 creation of highly localized excitations..." since this depends on the
 initial conditions. I suggest to write more carefully "... the pathway 
 to equipartition MAY LEAD to the creation of highly localized excitations..".

Author Ruffo :

We thank the referees for their careful reading of the text.

As explained below, we agree with many of the comments. However, some comments refer to too specific points from our point of view, and we are convinced that we should not go so deeply in a general article.

Thierry Dauxois and Stefano Ruffo

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Reply to Reviewer A

   This contribution needs significant rewriting, according to my
   understanding. Below are the points listed in the order of their
   appearance in the text.
   1. Caption Fig.1: Replace '/N' by '/(N+1)' in all denominators.

Yes. We modified the text accordingly.

   2. Section 'In real space: ...'
   Equation (2) is derived for periodic boundary conditions (PBC) as
   opposed to fixed boundary conditions (FBC) in the FPU studies.

It is correct and we have amended the text accordingly.

   Also it is just one of two (first order in time) equations, which are
   derived for right and left going waves. Both equations have to be
   combined in order to obtain the original field u_n. Therefore there
   is a more complicated transformation from w to u_n, it seems.

We agree (see also the reply to Referee D). We have modified the text. However, the derivation of the two KdV uncoupled equations for right-going and left-going waves is for us too detailed and not necessary for a Scholarpedia article. We have added a Reference.

    The legend in Fig.3 contains u, which can be easily misinterpreted
    as the original field u_n, while it should be w. I suggest to change
    the notations in the text, or to explain this point clearly in the caption.

We were already using w and not u in the legend in Fig. 3.

   It is interesting to note that exciting the k=1 mode in the FPU
   chain for weak nonlinearity alpha will NOT lead to the soliton train
   observed for a single KdV equation, but to a much smoother field
   wiggling in real space.

It's certainly correct, but we prefer to put the historical simulation by Zabusky on the PBC problem giving the KdV like behavior.

   It is the normal mode space (k) where the clear notion of an
   exponentially localized distribution of energies is observed.

We quote the exponential localization in the last section on "Recent Approaches" and it is definitely an interesting result. Again, we do not want to be too detailed here.

   Figure 3 is for alpha FPU, not beta FPU, as written in the caption.

Maybe, the Reviewer was meaning Fig. 4, where we were indeed wrong.

   Also it is taken from a recent publication by Zabusky et al. This
   should be corrected respectively cited.
   Also the initial condition in Fig.4 is NOT a normal mode, but a wave
   packet, which is quite localized in real space. Therefore it is not
   a pi mode. Also note, that a pi mode IS an exact normal mode for
   PBC, but is NOT for FBC.

We agree. Rather than describing separately Figs. 4 and 5, we have decided, for reasons of clarity, to erase Fig. 4, maintaining only one Figure for the description of the pi-mode evolution with PBC.

   Apparently in Fig.3 we see a simulation for PBC. This should be
   appropriately mentioned.

It was already mentioned in the previous version.

   Reading the review paper by Ford more carefully would show, that the
   FPU problem remained a mystery (at least to Ford) up to 1992. For
   good reasons, as explained there.

We have quoted Ford's review paper. This remark is also very mysterious for us!

   3. Section 'In normal mode space ...'
   First I think that this section should contain a much more detailed
   discussion of the various results on dynamics in normal mode space,
   which is right now squeezed into the section 'Recent approaches'.

This is not true, because the first section starts by discussing the FPU paradox in normal modes. It's true that there has been important recent work in normal mode space (the "two time scales theory", q-breathers, etc.) but it would not be appropriate to discuss this issues in the context of an historical account of the FPU problem. Maybe, this could be the matter for another Scholarpedia entry.

   These studies clearly show (and not just 'propose' that something
   'could') that the FPU trajectory DOES relax to equipartition. 

As Reviewer C mentions, this issue is not at all clarified, and it could well be that the FPU initial condition was chose in an ordered region of phase space, preventing the relaxation to equipartition. This this question is still debated, we prefer not to make strong statements.

   The issue is about at least two time scales - on a short one tau_1 a
   (typically exponentially) localized packet of energy distributions
   in mode space is reached (which is remarkably close to the profile
   when finding a periodic orbit, see Chaos 17, 023102 (2007) ). On a
   second, much larger scale tau_2 the system finally reaches
   equipartition. This scale tau_2 seems to depend sensitively on
   control parameters, and can be brought down to tau_1. Thus the FPU
   problem from a present perspective is: i) why are there two time
   scales in the relaxation? ii) what is the profile of the localized
   packet in mode space? iii) how does tau_2 depend on the control
   parameters?

As we have remarked above, although we agree that these are important issues (and indeed we briefly quoted the two-time-scales theory in the section on "Recent approaches"), we believe is not appropriate to expand it too much in this context.

   4. Section 'Relaxation to equipartition for short ...'
   This section is misleading. When repeating the FPU type experiment,
   for weak enough nonlinearities, one again observes a localized
   distribution in normal mode space up to some long time scale tau_2
   (e.g. Chaos 17, 023102 (2007) and references therein). Therefore
   there is no conceptual change here. However, when reaching the time
   scale tau_2, instead of equipartition, breathers in real space may
   form. I would suggest to clarify these issues. Breathers, or thermal
   equilibrium right after tau_2, is reached depending on short or long
   wavelength excitations. But the FPU problem (what happens BEFORE
   tau_2) is the same.

   5. Section 'Recent approaches'
   As already suggested above, most of
   these rather short and vague statements can be substantiated, and
   discussed in the above sections on normal mode space. This concerns
   also the results on periodic orbits, see
   http://www.pks.mpg.de/~flach/html/preprints_q_breathers.html , and
   in particular Physica D 237, 908 (2008), Am. J. Phys. 76, 453 (2008)
   and Int. J. Mod. Phys. B 21, 3925 (2007), where comparisons with
   resonant normal form analysis, and scaling properties of periodic
   orbits, are discussed.

In the "Instructions for authors" it is emphasized that "To be authoritative, the article should provide the full description of the topic, possibly ending with the details useful for experts." It is clearly what we have proposed.

It's clear that the referee wants to defend the recent and very interesting concept of q-breathers, but their relevance for the FPU relaxation problem is still debated and, from our point of view, they should be maintained in the last section we wrote. These papers are interesting but definitely belong to the 'Recent approaches' section.

We are convinced that the reader will look first for the definition of the FPU problem, and then for the two main (and old) approaches that are the solitons and the Resonance overlap criterion. Only experts will be interested in the others approached developed in the last few years. Anyway, we have added the Reference to Penati and Flach paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Reply to Reviewer C

  This is a very nice contribution, clear and well written. I agree 
  with the comments of the other reviewers and with the suggested changes.
  In particular, it is of fundamental importance that the FPU model DOES
  relax to equipartition, as reviewer A has remarked.
  However, at finite N it has been shown in: L. Casetti, M. Cerruti-Sola, 
  M. Pettini and E.G.D. Cohen, "The Fermi-Pasta-Ulam problem revisited: 
  Stochasticity thresholds in nonlinear Hamiltonian systems",Phys. Rev.E55, 
  6566 (1997), that a very steep drop of the largest Lyapunov exponent suggests 
  the existence of a stochasticity threshold (in principle a stochastic layer
  is always there, but its measure seems to be suddenly shrunk to a very small value).
  This stochasticity threshold vanishes, in energy density, as 1/N^2.
  In the section "In normal mode space: chaotic properties and Chirikov's 
  resonance overlap criterion", the authors mention the so-called
  strong stochasticity threshold, but the quotation: 
  Livi R. et al. (1985) "Equipartition threshold in nonlinear large Hamiltonian 
  systems: The Fermi-Pasta-Ulam model" Phys. Rev. A 31:1039-1045, is not
  appropriate there. The definition "strong stochasticity threshold" has been
  introduced for the first time in the papers: M. Pettini and M. Landolfi, 
  "Relaxation properties and ergodicity breaking in nonlinear hamiltonian dynamics",
  Phys. Rev. A41, 768 (1990), and M. Pettini and M. Cerruti-Sola, "Strong stochasticity 
  threshold in nonlinear large Hamiltonian systems: Effect on mixing times", 
  Phys. Rev. A44, 975 (1991),
  where a crossover in the energy density scaling of the largest Lyapunov exponent
  has been fist put in evidence, together with the fact that, in correspondence with
  this crossover, the relaxation time to equipartition of a nonequilibrium initial 
  condition also sharply changes its pattern (the relaxation times are steeply
  increasing by lowering the energy density below the crossover of the largest LE).
  I would ask the authors to emend the referencing according to this remark.

We have added a sentence and a quote to these papers.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Reply to Reviewer D

  The article is written in a very good way, however, there are points to be
  corrected. Apart from the technical misprints in the formulas and figure
  captions indicated by another Referee, I would add the following suggestions:
  1. I think it is better to interchange the figures 1 and 2, according to
  their appearance in the text.

We have modified the order of these two pictures.

  2. I suggest to indicate that the statistical behavior (relaxation
  to the steady-state distribution) was expected since the number
  of particle was quite large. Otherwise, for small number of particles
  one should not expect that the statistical mechanics is valid.
  For example, instead of
  "... contrary to the predictions of statistical mechanics, the energy..."
  it is better to write,
  "...contrary to the predictions of statistical mechanics for
  $N$ \goto \infty, ..." or, something like that.

We have introduced the modification "when the number of particles\(N\) is going to infinity"

3. The sentence:
"In 1965, using the so-called continuum limit, Zabusky and Kruskal
(Zabusky and Kruskal 1965) were able to explain the periodic ..."
seems to me a bit strong. I would say that more correct way is to say
"... were able to RELATE the periodic ...". I mean that the continuous
model where the soliton solution was found (first, numerically, and
after, analytically) is quite far from that in the discrete FPU model.
It is important that in the KdV model the interaction
between the waves traveling in opposite directions is absent, in
contrast with the original FPU model. It was shown numerically that
this interaction is crucial and directly related to the onset
of chaos (see details in the paper Comp. Phys. Comm. 5 (1973) 11-16).
In my opinion, one needs to indicate this point.

We have made the modification and introduced the reference.

4. When discussing the short wavelength limit, the impression is that the
dynamics in this case can be explained only in terms of breathers.
However, the approach based on the overlap of nonlinear resonances gives
a correct prediction for the chaos border in this case as well.
In particular, it seems to me not good to say that "... the pathway
to equipartition LEADS to the creation of highly localized excitations..."
since this depends on the initial conditions. I suggest to write more
carefully "... the pathway
to equipartition MAY LEAD to the creation of highly localized excitations..".

We have followed this suggestion. We have added a sentence, since indeed we had done ourselves this remark in the paper published in Chaos in 2005.

Reviewer A

Some of my previous comments have not been incorporated. I list them below (slightly modified, to may be give better understanding).

1. Section 'In real space: ...'

It is interesting to note that e.g. exciting the k=1 mode in the FPU chain for weak nonlinearity alpha will NOT lead to the soliton train observed for a single KdV equation, but to a much smoother field wiggeling in real space. It is the normal mode space (k) where the clear notion of an exponentially localized distribution of energies is observed. WITHOUT THIS INFORMATION, THE CHAPTER IS INCOMPLETE, AND THE READER MAY ERRONEOUSLY CONCLUDE, THAT SOLITONS IN REAL SPACE ARE THE CORRECT EXPLANATION OF THE FPU OBSERVATIONS. IN PARTICULAR, THE RECURRENCE PHENOMENON IS NOTHING BUT BEATING BETWEEN A FEW EXCITED DEGREES OF FREEDOM - WHATEVER THEIR NATURE. THEREFORE, THE LAST SENTENCE OF THAT SECTION IS MISLEADING AND PERHAPS EVEN WRONG. Now, I do not want the authors to change their taste, but I think it is fair to point exactly at the limits where the soliton picture disappears and breaks down altogether.

2. Section 'Recent approaches'

'It has been proposed that ... conditions could proceed ...' is wrong. Correct would be 'It has been observed ... conditions proceed ...'. These are reported facts, and not just suggestions.

understanding. Below are the points listed in the order of their appearance in the text.

1. Caption Fig.1

Replace '/N' by '/(N+1)' in all denominators (three times).

2. Section 'In real space: ...'

Equation (2) is derived for periodic boundary conditions (PBC) as opposed to fixed boundary conditions (FBC) in the FPU studies. Also it is just one of two (first order in time) equations, which are derived for right and left going waves. Both equations have to be combined in order to obtain the original field u_n.

Section 'In normal mode space ...'

First I think that this section should contain a much more detailed discussion of the various results on dynamics in normal mode space, which is right now squeezed into the section 'Recent approaches'. These studies clearly show (and not just 'propose' that something 'could') that the FPU trajectory DOES relax to equipartition. The issue is about at least two time scales - on a short one tau_1 a (typically exponentially) localized packet of energy distributions in mode space is reached (which is remarkably close to the profile when finding a periodic orbit, see Chaos 17, 023102 (2007) ). On a second, much larger scale tau_2 the system finally reaches equipartition. This scale tau_2 seems to depend sensitively on control parameters, and can be brought down to tau_1. Thus the FPU problem from a present perspective is: i) why are there two time scales in the relaxation? ii) what is the profile of the localized packet in mode space? iii) how does tau_2 depend on the control parameters?

Figure 3 is for alpha FPU, not beta FPU, as written in the caption. Also it is taken from a recent publication by Zabusky et al. This should be corrected respectively cited. Also the initial condition in Fig.4 is NOT a normal mode, but a wave packet, which is quite localized in real space. Therefore it is not a pi mode. Also note, that a pi mode IS an exact normal mode for PBC, but is NOT for FBC. Apparently in Fig.3 we see a simulation for PBC. This should be appropriately mentioned.

4. Section 'Relaxation to equipartition for short ...'

This section is misleading. When repeating the FPU type experiment, for weak enough nonlinearities, one again observes a localized distribution in normal mode space up to some long time scale tau_2 (e.g. Chaos 17, 023102 (2007) and references therein). Therefore there is no conceptual change here. However, when reaching the time scale tau_2, instead of equipartition, breathers in real space may form. I would suggest to clarify these issues. Breathers, or thermal equilibrium right after tau_2, is reached depending on short or long wavelength excitations. But the FPU problem (what happens BEFORE tau_2) is the same.

5. Section 'Recent approaches'

As already suggested above, most of these rather short and vague statements can be substantiated, and discussed in the above sections on normal mode space. This concerns also the results on periodic orbits, see http://www.pks.mpg.de/~flach/html/preprints_q_breathers.html , and in particular Physica D 237, 908 (2008), Am. J. Phys. 76, 453 (2008) and Int. J. Mod. Phys. B 21, 3925 (2007), where comparisons with resonant normal form analysis, and scaling properties of periodic orbits, are discussed.

======= Reviewer C ===========

This is a very nice contribution, clear and well written. I agree with the comments of the other reviewers and with the suggested changes. In particular, it is of fundamental importance that the FPU model DOES relax to equipartition, as reviewer A has remarked. However, at finite N it has been shown in: L. Casetti, M. Cerruti-Sola, M. Pettini and E.G.D. Cohen, "The Fermi-Pasta-Ulam problem revisited: Stochasticity thresholds in nonlinear Hamiltonian systems",Phys. Rev.E55, 6566 (1997), that a very steep drop of the largest Lyapunov exponent suggests the existence of a stochasticity threshold (in principle a stochastic layer is always there, but its measure seems to be suddenly shrunk to a very small value). This stochasticity threshold vanishes, in energy density, as 1/N^2.

In the section "In normal mode space: chaotic properties and Chirikov's resonance overlap criterion", the authors mention the so-called strong stochasticity threshold, but the quotation: Livi R. et al. (1985) "Equipartition threshold in nonlinear large Hamiltonian systems: The Fermi-Pasta-Ulam model" Phys. Rev. A 31:1039-1045, is not appropriate there. The definition "strong stochasticity threshold" has been introduced for the first time in the papers: M. Pettini and M. Landolfi, "Relaxation properties and ergodicity breaking in nonlinear hamiltonian dynamics", Phys. Rev. A41, 768 (1990), and M. Pettini and M. Cerruti-Sola, "Strong stochasticity threshold in nonlinear large Hamiltonian systems: Effect on mixing times", Phys. Rev. A44, 975 (1991), where a crossover in the energy density scaling of the largest Lyapunov exponent has been fist put in evidence, together with the fact that, in correspondence with this crossover, the relaxation time to equipartition of a nonequilibrium initial condition also sharply changes its pattern (the relaxation times are steeply increasing by lowering the energy density below the crossover of the largest LE).

I would ask the authors to emend the referencing according to this remark.

======= Reviewer D ===========

The article is written in a very good way, however, there are points to be corrected. Apart from the technical misprints in the formulas and figure captions indicated by another Referee, I would add the following suggestions:

1. I think it is better to interchange the figures 1 and 2, according to

  their appearance in the text.

2. I suggest to indicate that the statistical behavior (relaxation

 to the steady-state distribution) was expected since the number
 of particle was quite large. Otherwise, for small number of particles
one should not expect that the statistical mechanics is valid. 
 For example, instead of 

 "... contrary to the predictions of statistical mechanics, the energy..."

 it is better to write,

 "...contrary to the predictions of statistical mechanics for $N$ \goto \infty, ..."

 or, something like that.

3. The sentence:

 "In 1965, using the so-called continuum limit, Zabusky and Kruskal 
 (Zabusky and Kruskal 1965) were able to explain the periodic ..."
 seems to me a bit strong. I would say that more correct way is to say
 "... were able to RELATE the periodic ...". I mean that the continuous
 model where the soliton solution was found (first, numerically, and 
 after, analytically) is quite far from that in the discrete FPU model.  
 It is important that in the KdV model the interaction 
 between the waves traveling in opposite directions is absent, in 
 contrast with the original FPU model. It was shown numerically that 
 this interaction is crucial and directly related to the onset 
 of chaos (see details in the paper Comp. Phys. Comm. 5 (1973) 11-16). 
 In my opinion, one needs to indicate this point.

4. When discussing the short wavelength limit, the impression is that the dynamics

  in this case can be explained only in terms of breathers. However, the
 approach based on the overlap of nonlinear resonances gives a correct
 prediction for the chaos border in this case as well. In particular, it seems to me
 not good to say that "... the pathway to equipartition LEADS to the 
 creation of highly localized excitations..." since this depends on the
 initial conditions. I suggest to write more carefully "... the pathway 
 to equipartition MAY LEAD to the creation of highly localized excitations..".

Author Ruffo :

We thank the referees for their careful reading of the text.

As explained below, we agree with many of the comments. However, some comments refer to too specific points from our point of view, and we are convinced that we should not go so deeply in a general article.

Thierry Dauxois and Stefano Ruffo

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Reply to Reviewer A

   This contribution needs significant rewriting, according to my
   understanding. Below are the points listed in the order of their
   appearance in the text.
   1. Caption Fig.1: Replace '/N' by '/(N+1)' in all denominators.

Yes. We modified the text accordingly.

   2. Section 'In real space: ...'
   Equation (2) is derived for periodic boundary conditions (PBC) as
   opposed to fixed boundary conditions (FBC) in the FPU studies.

It is correct and we have amended the text accordingly.

   Also it is just one of two (first order in time) equations, which are
   derived for right and left going waves. Both equations have to be
   combined in order to obtain the original field u_n. Therefore there
   is a more complicated transformation from w to u_n, it seems.

We agree (see also the reply to Referee D). We have modified the text. However, the derivation of the two KdV uncoupled equations for right-going and left-going waves is for us too detailed and not necessary for a Scholarpedia article. We have added a Reference.

    The legend in Fig.3 contains u, which can be easily misinterpreted
    as the original field u_n, while it should be w. I suggest to change
    the notations in the text, or to explain this point clearly in the caption.

We were already using w and not u in the legend in Fig. 3.

   It is interesting to note that exciting the k=1 mode in the FPU
   chain for weak nonlinearity alpha will NOT lead to the soliton train
   observed for a single KdV equation, but to a much smoother field
   wiggling in real space.

It's certainly correct, but we prefer to put the historical simulation by Zabusky on the PBC problem giving the KdV like behavior.

   It is the normal mode space (k) where the clear notion of an
   exponentially localized distribution of energies is observed.

We quote the exponential localization in the last section on "Recent Approaches" and it is definitely an interesting result. Again, we do not want to be too detailed here.

   Figure 3 is for alpha FPU, not beta FPU, as written in the caption.

Maybe, the Reviewer was meaning Fig. 4, where we were indeed wrong.

   Also it is taken from a recent publication by Zabusky et al. This
   should be corrected respectively cited.
   Also the initial condition in Fig.4 is NOT a normal mode, but a wave
   packet, which is quite localized in real space. Therefore it is not
   a pi mode. Also note, that a pi mode IS an exact normal mode for
   PBC, but is NOT for FBC.

We agree. Rather than describing separately Figs. 4 and 5, we have decided, for reasons of clarity, to erase Fig. 4, maintaining only one Figure for the description of the pi-mode evolution with PBC.

   Apparently in Fig.3 we see a simulation for PBC. This should be
   appropriately mentioned.

It was already mentioned in the previous version.

   Reading the review paper by Ford more carefully would show, that the
   FPU problem remained a mystery (at least to Ford) up to 1992. For
   good reasons, as explained there.

We have quoted Ford's review paper. This remark is also very mysterious for us!

   3. Section 'In normal mode space ...'
   First I think that this section should contain a much more detailed
   discussion of the various results on dynamics in normal mode space,
   which is right now squeezed into the section 'Recent approaches'.

This is not true, because the first section starts by discussing the FPU paradox in normal modes. It's true that there has been important recent work in normal mode space (the "two time scales theory", q-breathers, etc.) but it would not be appropriate to discuss this issues in the context of an historical account of the FPU problem. Maybe, this could be the matter for another Scholarpedia entry.

   These studies clearly show (and not just 'propose' that something
   'could') that the FPU trajectory DOES relax to equipartition. 

As Reviewer C mentions, this issue is not at all clarified, and it could well be that the FPU initial condition was chose in an ordered region of phase space, preventing the relaxation to equipartition. This this question is still debated, we prefer not to make strong statements.

   The issue is about at least two time scales - on a short one tau_1 a
   (typically exponentially) localized packet of energy distributions
   in mode space is reached (which is remarkably close to the profile
   when finding a periodic orbit, see Chaos 17, 023102 (2007) ). On a
   second, much larger scale tau_2 the system finally reaches
   equipartition. This scale tau_2 seems to depend sensitively on
   control parameters, and can be brought down to tau_1. Thus the FPU
   problem from a present perspective is: i) why are there two time
   scales in the relaxation? ii) what is the profile of the localized
   packet in mode space? iii) how does tau_2 depend on the control
   parameters?

As we have remarked above, although we agree that these are important issues (and indeed we briefly quoted the two-time-scales theory in the section on "Recent approaches"), we believe is not appropriate to expand it too much in this context.

   4. Section 'Relaxation to equipartition for short ...'
   This section is misleading. When repeating the FPU type experiment,
   for weak enough nonlinearities, one again observes a localized
   distribution in normal mode space up to some long time scale tau_2
   (e.g. Chaos 17, 023102 (2007) and references therein). Therefore
   there is no conceptual change here. However, when reaching the time
   scale tau_2, instead of equipartition, breathers in real space may
   form. I would suggest to clarify these issues. Breathers, or thermal
   equilibrium right after tau_2, is reached depending on short or long
   wavelength excitations. But the FPU problem (what happens BEFORE
   tau_2) is the same.

   5. Section 'Recent approaches'
   As already suggested above, most of
   these rather short and vague statements can be substantiated, and
   discussed in the above sections on normal mode space. This concerns
   also the results on periodic orbits, see
   http://www.pks.mpg.de/~flach/html/preprints_q_breathers.html , and
   in particular Physica D 237, 908 (2008), Am. J. Phys. 76, 453 (2008)
   and Int. J. Mod. Phys. B 21, 3925 (2007), where comparisons with
   resonant normal form analysis, and scaling properties of periodic
   orbits, are discussed.

In the "Instructions for authors" it is emphasized that "To be authoritative, the article should provide the full description of the topic, possibly ending with the details useful for experts." It is clearly what we have proposed.

It's clear that the referee wants to defend the recent and very interesting concept of q-breathers, but their relevance for the FPU relaxation problem is still debated and, from our point of view, they should be maintained in the last section we wrote. These papers are interesting but definitely belong to the 'Recent approaches' section.

We are convinced that the reader will look first for the definition of the FPU problem, and then for the two main (and old) approaches that are the solitons and the Resonance overlap criterion. Only experts will be interested in the others approached developed in the last few years. Anyway, we have added the Reference to Penati and Flach paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Reply to Reviewer C

  This is a very nice contribution, clear and well written. I agree 
  with the comments of the other reviewers and with the suggested changes.
  In particular, it is of fundamental importance that the FPU model DOES
  relax to equipartition, as reviewer A has remarked.
  However, at finite N it has been shown in: L. Casetti, M. Cerruti-Sola, 
  M. Pettini and E.G.D. Cohen, "The Fermi-Pasta-Ulam problem revisited: 
  Stochasticity thresholds in nonlinear Hamiltonian systems",Phys. Rev.E55, 
  6566 (1997), that a very steep drop of the largest Lyapunov exponent suggests 
  the existence of a stochasticity threshold (in principle a stochastic layer
  is always there, but its measure seems to be suddenly shrunk to a very small value).
  This stochasticity threshold vanishes, in energy density, as 1/N^2.
  In the section "In normal mode space: chaotic properties and Chirikov's 
  resonance overlap criterion", the authors mention the so-called
  strong stochasticity threshold, but the quotation: 
  Livi R. et al. (1985) "Equipartition threshold in nonlinear large Hamiltonian 
  systems: The Fermi-Pasta-Ulam model" Phys. Rev. A 31:1039-1045, is not
  appropriate there. The definition "strong stochasticity threshold" has been
  introduced for the first time in the papers: M. Pettini and M. Landolfi, 
  "Relaxation properties and ergodicity breaking in nonlinear hamiltonian dynamics",
  Phys. Rev. A41, 768 (1990), and M. Pettini and M. Cerruti-Sola, "Strong stochasticity 
  threshold in nonlinear large Hamiltonian systems: Effect on mixing times", 
  Phys. Rev. A44, 975 (1991),
  where a crossover in the energy density scaling of the largest Lyapunov exponent
  has been fist put in evidence, together with the fact that, in correspondence with
  this crossover, the relaxation time to equipartition of a nonequilibrium initial 
  condition also sharply changes its pattern (the relaxation times are steeply
  increasing by lowering the energy density below the crossover of the largest LE).
  I would ask the authors to emend the referencing according to this remark.

We have added a sentence and a quote to these papers.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Reply to Reviewer D

  The article is written in a very good way, however, there are points to be
  corrected. Apart from the technical misprints in the formulas and figure
  captions indicated by another Referee, I would add the following suggestions:
  1. I think it is better to interchange the figures 1 and 2, according to
  their appearance in the text.

We have modified the order of these two pictures.

  2. I suggest to indicate that the statistical behavior (relaxation
  to the steady-state distribution) was expected since the number
  of particle was quite large. Otherwise, for small number of particles
  one should not expect that the statistical mechanics is valid.
  For example, instead of
  "... contrary to the predictions of statistical mechanics, the energy..."
  it is better to write,
  "...contrary to the predictions of statistical mechanics for
  $N$ \goto \infty, ..." or, something like that.

We have introduced the modification "when the number of particles\(N\) is going to infinity"

3. The sentence:
"In 1965, using the so-called continuum limit, Zabusky and Kruskal
(Zabusky and Kruskal 1965) were able to explain the periodic ..."
seems to me a bit strong. I would say that more correct way is to say
"... were able to RELATE the periodic ...". I mean that the continuous
model where the soliton solution was found (first, numerically, and
after, analytically) is quite far from that in the discrete FPU model.
It is important that in the KdV model the interaction
between the waves traveling in opposite directions is absent, in
contrast with the original FPU model. It was shown numerically that
this interaction is crucial and directly related to the onset
of chaos (see details in the paper Comp. Phys. Comm. 5 (1973) 11-16).
In my opinion, one needs to indicate this point.

We have made the modification and introduced the reference.

4. When discussing the short wavelength limit, the impression is that the
dynamics in this case can be explained only in terms of breathers.
However, the approach based on the overlap of nonlinear resonances gives
a correct prediction for the chaos border in this case as well.
In particular, it seems to me not good to say that "... the pathway
to equipartition LEADS to the creation of highly localized excitations..."
since this depends on the initial conditions. I suggest to write more
carefully "... the pathway
to equipartition MAY LEAD to the creation of highly localized excitations..".

We have followed this suggestion. We have added a sentence, since indeed we had done ourselves this remark in the paper published in Chaos in 2005.

Reviewer A

Some of my previous comments have not been incorporated. I list them below (slightly modified, to may be give better understanding).

1. Section 'In real space: ...'

It is interesting to note that e.g. exciting the k=1 mode in the FPU chain for weak nonlinearity alpha will NOT lead to the soliton train observed for a single KdV equation, but to a much smoother field wiggeling in real space. It is the normal mode space (k) where the clear notion of an exponentially localized distribution of energies is observed. WITHOUT THIS INFORMATION, THE CHAPTER IS INCOMPLETE, AND THE READER MAY ERRONEOUSLY CONCLUDE, THAT SOLITONS IN REAL SPACE ARE THE CORRECT EXPLANATION OF THE FPU OBSERVATIONS. IN PARTICULAR, THE RECURRENCE PHENOMENON IS NOTHING BUT BEATING BETWEEN A FEW EXCITED DEGREES OF FREEDOM - WHATEVER THEIR NATURE. THEREFORE, THE LAST SENTENCE OF THAT SECTION IS MISLEADING AND PERHAPS EVEN WRONG. Now, I do not want the authors to change their taste, but I think it is fair to point exactly at the limits where the soliton picture disappears and breaks down altogether.

2. Section 'Recent approaches'

'It has been proposed that ... conditions could proceed ...' is wrong. Correct would be 'It has been observed ... conditions proceed ...'. These are reported facts, and not just suggestions.

Reply to Reviewer A

 Some of my previous comments have not been incorporated.
 I list them below (slightly modified, to may be give better understanding).

 1. Section 'In real space: ...'

 It is interesting to note that e.g. exciting the k=1 mode in the FPU chain
 for weak nonlinearity alpha will NOT lead to the soliton train observed
 for a single KdV equation, but to a much smoother field wiggeling
 in real space. It is the normal mode space (k) where the clear notion
 of an exponentially localized distribution of energies is observed.
 WITHOUT THIS INFORMATION, THE CHAPTER IS INCOMPLETE, AND THE READER MAY
 ERRONEOUSLY CONCLUDE, THAT SOLITONS IN REAL SPACE ARE THE CORRECT
 EXPLANATION OF THE FPU OBSERVATIONS. IN PARTICULAR, THE RECURRENCE PHENOMENON
 IS NOTHING BUT BEATING BETWEEN A FEW EXCITED DEGREES OF FREEDOM - WHATEVER
 THEIR NATURE. THEREFORE, THE LAST SENTENCE OF THAT SECTION IS MISLEADING
 AND PERHAPS EVEN WRONG. Now, I do not want the authors to change their
 taste, but I think it is fair to point exactly at the limits where the
 soliton picture disappears and breaks down altogether.

The reason why a small amplitude initial condition of the original FPU problem with fixed boundary consitions might not lead to the formation of a soliton train could be twofold. On one hand this could be due to the interaction between right and left going nonlinear waves, on the other if the nonlinearity is not strong enough, the dispersion term will win and make the emergence of solitons hard because of the weakness of the shock wave mechanism. To take into account the referee's remark, we have added a sentence pointing to the necessity of having enough nonlinearity.

Moreover, we agree that mode space beating is certainly an alternative way of looking at the recurrence phenomenon. However, we are not aware of quantitative estimates of the recurrence time using this method. On the contrary, this exists using soliton theory (e.g. Toda estimate). We had quoted before Jackson's work devoted to the study of recurrence time in mode space in the Section on normal modes. Anyway, we have decided to remove in the last sentence of the Section on real space any reference to normal modes approaches, which might well in the future produce similar estimates.

   2. Section 'Recent approaches'

  'It has been proposed that ... conditions could proceed ...' is wrong.
  Correct would be 'It has been observed ... conditions proceed ...'.
  These are reported facts, and not just suggestions.

We have changed the sentence as asked.