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Chesterfield Canal

The Chesterfield Canal is a narrow canal in the East Midlands of England and it is known locally as 'Cuckoo Dyke'. It was one of the last of the canals designed by James Brindley, who died while it was being constructed. It was opened in 1777 and ran for 46 mi from the River Trent at West Stockwith, Nottinghamshire to Chesterfield, Derbyshire, passing through the Norwood Tunnel at Kiveton Park, at the time one of the longest tunnels on the British canal system. The canal was built to export coal, limestone, and lead from Derbyshire, iron from Chesterfield, and corn, deals, timber, groceries and general merchandise into Derbyshire. The stone for the Palace of Westminster was quarried in North Anston, Rotherham, and transported via the canal.

It was reasonably profitable, paying dividends from 1789, and with the coming of the railways, some of the proprietors formed a railway company. It became part of the Manchester, Sheffield and Lincolnshire Railway company, and although there were intermittent plans to convert parts of it to a railway, it continued to thrive as a canal. In 1907, subsidence from local coal mines caused the collapse of the Norwood Tunnel, and the canal was effectively split in two. Subsequently, the main use of the Chesterfield end was the supply of water to the iron industry, while commercial carrying continued on the Worksop to West Stockwith section until the late 1950s.

It was formally closed in 1961, but campaigners fought for it to be retained, and the Worksop to Stockwith section was designated as a "cruiseway" under the Transport Act 1968, meaning that it would be retained for leisure use. The rest was designated as a remainder waterway, and parts were sold off, with housing being built over the route through Killamarsh. The Chesterfield Canal Society was formed in 1978 to spearhead restoration, becoming the Chesterfield Canal Trust in 1997. They initially sought to extend the navigable section beyond Worksop, but when progress was slow, moved to working on the Chesterfield end. Over 5 mi of canal, including five original locks and a brand new lock at Staveley Basin were navigable by 2017. The eastern end was restored from Worksop to the mouth of the Norwood Tunnel at Kiveton Park near Rotherham, South Yorkshire, between 1995 and 2003, funded by Derelict Land Grants, English Partnerships and the Heritage Lottery Fund.

Less than 9 mi of the original route remain to be restored to link the two navigable sections, but this will require some new lengths of canal to be built, to bypass the housing development at Killamarsh, and to replace most of the Norwood Tunnel, which cannot be restored. The eastern section is managed by the Canal and River Trust, while the western section is managed by Derbyshire County Council. It includes Tapton Lock Visitor Centre, located to the north of Tapton Park, and the Hollingwood Hub, which provides offices for the Trust, together with meeting rooms and a cafe. It is located by Hollingwood Lock, and consists of a large new extension on the back of the refurbished lock house.

History
Lead had been mined in Derbyshire on a commercial scale since at least 1300, but the industry was hampered by poor transport links. The pigs of lead were carried by pack horses to Bawtry, where there were wharves on the River Idle. Small boats carried them to the River Trent, where the cargo was transferred to larger vessels for onward distribution. There was no proper system of road maintenance, and so the use of carts and wagons for the over-land journey was not possible. The River Idle was also unreliable, being affected by floods and droughts, but there were no alternatives, and the industry flourished despite the difficulties. The Derwent Navigation was authorised in 1719, running from Derby to the River Trent, but it was less reliable than the Idle, and so did not seriously affect trade through Bawtry. The London Lead Company, one of the main traders, together with others involved in lead and coal mining, looked for possible solutions, and were joined in this by several prominent landowners.

The main impetus for a canal came from the owners of mineral rights. The London Lead Company met with the Duke of Devonshire, the Duke of Newcastle, and others, but proposals were actively supported by the towns of Chesterfield, Worksop and Retford. Retford had obtained an Act of Parliament in 1720 to improve the River Idle from Bawtry to the town, but very little work was done to implement the scheme. In addition to being a market town, Retford also manufactured bricks and tiles, had reserves of gravel, and produced an agricultural surplus. In 1769, a steering group produced a pamphlet extolling the advantages of a proposed canal, which was available to everyone, but was primarily addressed to the Duke of Newcastle, Lord George Augustus Cavendish, and the Member of Parliament (MP) for Derbyshire, Godfrey Bagnall Clarke. The route of the canal was surveyed by James Brindley and John Varley, who estimated the cost at £94,908 17s. Brindley presented his proposals to a meeting in Worksop on 24 August 1769. Bawtry interests asked John Grundy to carry out a second survey. He proposed a rather shorter course, from Stockwith in a straight line to Bawtry and then by Scrooby, Blyth and Carlton, to join Brindley's line at Shireoaks. Grundy's line was 5+1/2 mi shorter, and the cost estimated at £71,479 6s $9 1/2$d. Although Grundy's line was considerably cheaper, it achieved this by missing Worksop and Retford, and the investors had already decided in favour of Brindley's route. An application was made to Parliament and the Act of Parliament received the Royal Assent on 28 March 1771, entitled An Act for making a navigable Cut or Canal from Chesterfield, in the county of Derby, through or near Worksop and Retford, to join the River Trent, at or near Stockwith, in the county of Nottingham.

The promoters consisted of one hundred and seventy-four persons, amongst whom were the Duke of Devonshire, the Duke of Newcastle, Lord Scarsdale, the Dean of York, and Sir Cecil Wray. They were incorporated by the name of The Company of Proprietors of the Canal Navigation from Chesterfield to the River Trent, and empowered to raise among themselves the sum of £100,000, in one thousand shares of £100 each, to fund the construction.

Immediately on the passing of the act, construction began under the direction of Brindley. Upon his death in September 1772, John Varley moved from Clerk of Works to Resident Engineer with Hugh Henshall, Brindley's brother-in-law, appointed in a consultative capacity. He agreed to make inspections every three months. Henshall became Chief Engineer in 1773 and John Varley continued to work on the project. By August 1773, the section between Shireoaks and Worksop was nearly complete, and the company created a works to build boats at Shireoaks. The boats were used to carry spoil from the construction, and also to deliver coal. The canal was effectively open from Shireoaks to below Worksop by 6 April 1774, and to Retford by 3 August. The price of coal to the residents of Retford dropped from 15s 6d (77.5p) to 10s 6d (52.5p) overnight. Hayton was reached on 22 February 1775, and the Norwood Tunnel opened on 9 May 1775. The canal was to be built as a narrow canal, but in 1775, nine shareholders offered to fund the extra cost of making it a broad canal from Retford to Stockwith. Retford Corporation joined them, and each contributed £500, although the additional cost exceeded the £6,000 subscribed. The canal was opened throughout on 4 June 1777, with the lock into the River Trent at West Stockwith completed in the autumn, but the only record of wide-beamed boats using it at Retford is prior to 1799.

Operation
As built, the canal was almost 46 mi long, being 25 mi from the Trent to Worksop with a rise of 95 ft. From Worksop to the eastern portal of Norwood Tunnel it was 6.1 mi with a further 145 ft rise. From there to Chesterfield it was a further 13.9 mi with a fall of 73 ft followed by a rise of 40 ft. There were 65 locks in all, with two tunnels: a short 154 yd tunnel near Gringley Beacon, and the major 2884 yd long Norwood Tunnel. At the time of construction, Norwood Tunnel was the second longest canal tunnel in Britain, exceeded only by another of Brindley's tunnels, that at Harecastle on the Trent and Mersey Canal. The canal was a typical Brindley contour canal, following the contours to avoid costly cuttings and embankments, which resulted in a less than direct route in places.

In late 1775, the company had decided to build a branch, which left the canal between Renishaw and Staveley, and ran generally southwards for about 1 mi to the turnpike road at Norbriggs. This may have been to help with transport to Chesterfield, until the main line was completed, but in 1777, the company leased Norbriggs Colliery. Henshall supervised its operation, and transport of coal to the branch canal was improved by the provision of a "Newcastle Rail Road" in 1778, which used wooden rails. The company pulled out of running the mine in 1797, when they advertised it to be let. Water supply was initially by a reservoir at Pebley, which was later supplemented by reservoirs at Harthill, Woodall and Killamarsh. Near Worksop, a private branch was built to serve the Lady Lee quarry, which ran for about 0.75 mi, while at Netherthorpe, a branch connected to the East Inkersall tramroad, which served pits near the Adelphi Canal. A gasworks was later built at the junction, and while the 1877 Ordnance survey map shows a short stub of the branch, connecting to the Seymour and Speedwell Branch railway, by 1898 a railway siding had replaced it, leaving just a basin at the junction to serve the gasworks. Another private branch was built in 1840, which crossed the Norbriggs road at Netherthorpe to serve the Speedwell Colliery. It was built by the Barrows, who also owned Staveley ironworks, but by 1860 had been replaced by a railway.

By 1789, the company had spent a total of £152,400, raised through shares and by mortgages. Income for the year was £8,320, and after interest payments had been met, that left a net profit of £2,780, enabling them to make their first dividend payment of one per cent. 74,312 tons of traffic were carried that year, of which 42,380 tons were coal, with the rest consisting of lead, iron, stone, corn, lime, timber and sundries. Dividends had risen to five per cent by 1795, were eight per cent the following year, and were six or seven per cent until at least 1836. There were several suggestions for links to the River Don Navigation or the Sheffield Canal, the first in 1792 as part of the scheme to build a canal to Sheffield, and subsequently in 1824, as part of the Grand Commercial Canal, and in 1852, when a Sheffield and Chesterfield Junction Canal was proposed, but none of them moved past the planning stage.

Railway age
In the 1840s, Railway Mania was gripping the country, with many speculative schemes being proposed. Rather than face steady demise as the new form of transport developed, a number of the proprietors formed the Manchester & Lincoln Union Railway company in October 1845. The intention was to build a line from Liverpool to Grimsby, and to convert part of the canal into a railway. Locally, this included a line from Staveley to Gainsborough, via Worksop, with a branch to Lincoln and another from Worksop to Beighton. The previous year, the Sheffield & Lincolnshire Junction Railway had been proposed, to build a line from Sheffield to Gainsborough, and was supported by the Sheffield, Ashton-under-Lyne & Manchester Railway. The new scheme was in direct competition to the earlier one, but the two sets of proprietors met and agreed to amalgamate. Now arguing from a position of strength, the Manchester & Lincoln Union rescinded their plans for most of their lines, and obtained an Act of Parliament on 7 August 1846 allowing them to construct a line from the Midland Railway at Staveley to the canal at Worksop. The new company amalgamated with the Chesterfield Canal, to become the rather unwieldy Manchester & Lincoln Union Railway & Chesterfield & Gainsborough Canal, with powers to further amalgamate with the Sheffield & Lincolnshire Junction Railway. However, by the time they announced this, the Sheffield & Lincolnshire Junction Railway had already become part of the Manchester, Sheffield and Lincolnshire Railway, and so they amalgamated with that company instead. The Act of Parliament stated that although the canal company was to be dissolved, the entire canal was to be retained, its water supplies maintained, and tolls were to be fixed at a just level.

Under the new regime, the canal was put back into good order in 1848, after many years where regular maintenance had been neglected. The benefits were immediate, with traffic increasing as a result. Over 200,000 tons were carried in 1848, the highest recorded total. The railway company started to operate as a carrier on the canal, and to maintain water levels, the reservoirs at Chesterfield were enlarged in 1856. Traffic levels were holding up in 1858, despite fierce competition from the London and North Western Railway. However, the owners considered options for converting the canal from Kiverton Park to Chesterfield into a railway in 1872, 1873 and 1884, but on each occasion, the plans were shelved. The Manchester, Sheffield and Lincolnshire Railway was expanding, and an Act of Parliament was obtained in 1889, which allowed them to divert parts of the canal to enable a new railway connecting Chesterfield, Staveley and Beighton to be constructed. The new route was straighter than the old, and reduced the length of the canal by around 1/2 mi. The Beighton to Staveley section of the new railway opened on 1 December 1891, and reached Chesterfield on 4 June 1892. The railway's carrying business on this and other canals ceased shortly afterwards. The original route of the canal at Hague Lane was largely destroyed by the construction of the railway and Renishaw Central station, but further north, the section around Chapel Wheel Forge and Dam is still clearly visible, and reinstatement of it as part of a restored canal was considered but rejected on practical grounds. Close to Bluebank Lock, the railway, known as the Great Central Railway Chesterfield Loop, turned to the east and passed to the south of Staveley Iron Works to reach Staveley Town station. Prior to construction of this section, the canal looped to the north around the iron works, passing through Cinderhill Lock before turning south-eastwards. The canal was re-routed along the southern edge of the works, and the railway crossed over at Staveley Works station, almost on top of a new lock which was known as Staveley Works Lock or Hollingwood Lock.

Parts of the canal above Worksop were affected by subsidence from local coal mines. By 1905, traffic had dropped to 45,177 tons, of which around 15,000 tons were coal and 11,000 were bricks. Some 40 boats were still working on the canal, although a short section between Staveley and Chesterfield had become unnavigable. The canal was making a loss, with receipts of £1,837 and expenditure of £3,883 in 1905. The largest drain on resources was the Norwood Tunnel, where £21,000 had been spent since 1871, in an attempt to repair damage and raise the roof. However, on 18 October 1907 there was a further substantial collapse, and the tunnel was closed. Effectively, the canal above Worksop became redundant. Responsibility for the canal passed the London and North Eastern Railway (LNER) in 1923, who carried out regular maintenance, and repaired and enlarged the lock connecting West Stockwith basin to the River Trent between 1923 and 1925. Various attempts to reduce the growth of weed, which had first appeared on the canal in 1852, were made, with variable success, and there was a brisk trade in munitions during the Second World War. The last serious commercial traffic on the canal was the carriage of bricks from Walkeringham to West Stockwith, which ended in 1955, although there was a small trade in warp, a fine silt dredged from the Trent at the mouth of the River Idle, and used by the cutlery trade in Sheffield for polishing metal, which lasted into the early 1960s. Parts of the isolated section from the tunnel to Chesterfield were infilled and redeveloped.

Restoration
Following the post-war Labour Government nationalisation of the railways and the canals, the Chesterfield Canal became the responsibility of the British Transport Commission. They were determined to close any canals that were not used to carry freight, and did not see any future in leisure use. By 1961, the canal had been formally closed, all requests to use it for leisure purposes were refused, and there were proposals for official abandonment of its entire length. Much of it was filled with rubbish and stagnant water, but despite this, Cliff Clarke, a businessman from South Yorkshire, mounted a campaign to save it. The British Transport Commission agreed to keep the canal open for two more years in 1962, although the lockgates from Worksop to Kiveton were removed, and the lock walls bulldozed, on the grounds of public safety. Clarke formed the Retford and Worksop (Chesterfield Canal) Boat Club soon afterwards, and it became a limited company in 1966, to cope with the growing membership. Unofficial work parties were organised, to clear locks and cut back vegetation and weed; they were unofficial because the British Transport Commission would not allow volunteers to work on their property. Such activity became slightly more collaborative when the canal became the responsibility of the British Waterways Board. Government policy changed following the publication of a White Paper entitled British Waterways: Recreation and Amenity in September 1967, and the canal was classified as a cruiseway between the Trent and Morse Lock, Worksop under the Transport Act 1968, meaning that it would be maintained to a navigable standard for leisure use. The rest of the canal was classified as remainder.

By 1968, restoration efforts were sufficiently organised that the Chesterfield Canal was mentioned in the May edition of the Bulletin, produced for members of the Inland Waterways Association (IWA) by the forerunners of the Waterway Recovery Group. The Chesterfield Canal Society was formed in September 1976, determined to restore the canal beyond Worksop, and in 1977 they organised a boat rally at Worksop to commemorate the 200th anniversary of the canal, in conjunction with the Worksop and Retford Boat Club. Bassetlaw District Council provided financial backing, and the event was held over four days, coinciding with the Queen's Silver Jubilee. Some 150 boats and over 21,000 people attended, with catering services, theatrical performances and exhibitions held on the Bracebridge showground. The Duke of Devonshire opened the festivities, and the Chesterfield Canal Society was formally launched, with a view to attracting support from anyone interested in the canal, rather than just from boaters.

Progress beyond Worksop moved forwards rapidly in 1995, when a Derelict Land Grant was obtained by Nottinghamshire County Council, and with additional funding from British Coal, a £4.5 million scheme began to restore the canal between Worksop and the aqueduct over the River Ryton which forms the county boundary. Rotherham Borough Council were hoping to obtain similar grants for the section between the aqueduct and the Norwood Tunnel, but the funding came from the Heritage Lottery Fund and the regeneration agency English Partnerships. The complete project cost £19 million, which included an extra £971,000 received from the Heritage Lottery Fund, to allow detailed archaeological investigation of the Thorpe and Turnerwood lock flights to be undertaken. On 18 November 2002, the first boats in over 70 years ascended the Turnerwood flight. Thirty locks had been restored, from Morse Lock (49) near Worksop to Thorpe Top Treble Lock (20) near Thorpe Salvin. Three bridges had been rebuilt with navigable headroom, one over Tylden Road, the main access road into Rhodesia, an adjacent one that provided access to nearby cottages, and one which carried Shireoaks Common over the canal near to Shireoaks railway station. Just before Ryton Aqueduct, a new shallow lock named Boundary Lock (41a) had been required, due to widespread subsidence affecting the ground levels, and at Shireoaks, the former colliery loading basin had been turned into a marina.

Western section
Restoration had focused initially on the route east from Norwood Tunnel to Worksop, which presented relatively few physical obstacles to being restored to navigation. However, in practice progress had been slow, and attention turned to the section west of Norwood Tunnel, where much more damage had been done to the canal bed, with it being filled in and built over in many places. This included the construction of housing at Killamarsh in the 1970s, which went ahead despite protests. However, parts of the channel had survived remarkably well in places, as it had been used to supply water to the iron industry, and maintained by a team with a boat until the 1950s. The last few miles of the canal, from Chesterfield to Staveley, were in reasonable condition, although the towpath was overgrown and difficult to access, while much of the route was under threat from opencast coal mining and a planned bypass, which had first been proposed in 1927. Regular workparties began the process of restoration in 1988, organised by the Chesterfield Canal Society, and supplemented by volunteers from the Waterway Recovery Group periodically. Tapton lock was the first to be reopened, in 1990. Derbyshire County Council acquired the section of canal between Chesterfield and Staveley and secured derelict land grants to enable dredging and towpath maintenance to take place. A campaign to ensure that once all the coal had been extracted from the opencast mine, the canal would be reinstated was successful. Two lowered bridges had been rebuilt by June 2001, and Cow Lane Bridge followed in May 2002. Four more locks were restored, with the 5 mi section from Chesterfield being reopened to navigation in 2002.

In a separate development, a private owner of a length of the canal near Boiley Farm, Killamarsh, obtained a Derelict Land Grant to enable him to restore around 550 yd of the channel in 1992. Although used as fishing ponds, the work resulted in a navigable profile being re-established, and a water supply was obtained from a small unnamed brook, which feeds into the southern end of the ponds. An earth bund at the site of the former Gallas footbridge divides the channel into two ponds, and carries a public footpath. In 1997, the Chesterfield Canal Trust was formed, as a limited company with charitable aims, and took over the assets of the Canal Society in 1998.

A breach in the canal in March 2007, caused by the collapse of a culvert carrying the Tinkersick Brook under the canal, forced temporary closure of all but the top pound above Tapton Lock for about six weeks. Near the site of the former Renishaw Iron Foundry, which closed in 1992 and was subsequently redeveloped for housing, a length of canal was reexcavated in 2007–08. The work included the new Renishaw Foundry footbridge (18a), which connects the housing to green space and a play area on the bank of the canal, and deep piling where an embankment originally carried the canal over the Smithy Brook. The development of the site ceased in 2010, when there were issues with ownership of the land. Since then, there have been issues with vandalism, and the lack of a natural water supply has been a continuing problem.

Over the winter of 2010–11, the Canal Trust obtained its first permanent base, when the lock house at Hollingwood Lock was renovated and a large extension added to the rear. The building, which is known as the Hollingwood Hub, was funded by a grant from Community Assets, part of the government's Office for Civil Society, and provides offices for the Trust, a meeting room and a coffee shop. The ecology of the area was improved by the planting of 450 trees, provided by the Woodland Trust, in November 2011.

The next big advance was the opening of some 660 yd of canal beyond Mill Green Bridge, and the construction of a new mooring basin at Staveley, which was completed in early 2012. Funding for the basin was provided by Derbyshire County Council with a grant from the East Midlands Development Agency, and the work which included the reclamation of the surrounding land, won an award from the Institute of Civil Engineers, given jointly to Derbyshire and the contractors NT Killingley. The work was aided by a road-building scheme for a new Staveley Bypass, which enabled two bridges to be raised and a third to be built, all with sufficient clearance for navigation.

North from Staveley
Progress beyond Staveley Basin was hampered by a railway bridge. The first bridge over the canal in this location was further to the east, and was erected in the 1850s, when the canal arm to Speedwell Colliery was filled in, to be converted to a railway. The Staveley Ironworks railway system was sold to the Midland Railway in the 1870s, who built a replacement line and bridge, to ease the sharpness of the curve. This area has suffered from subsidence, so when the canal was formally abandoned, the twin-track iron bow girder bridge was removed and the canal was filled in. A combination of the subsidence, and the use of a deep ballast base to enable tracks to be carried straight over any new bridge has reduced the available headroom by 2.25 m. The solution adopted to overcome this was the construction of a dropped pound, with a new Staveley Town Lock (5a) immediately to the north of Staveley Basin to lower the level of the canal, and Railway Lock (5b) constructed just beyond the railway bridge. A syphon will maintain the water supply to the canal beyond the dropped pound. Staveley Town Lock was completed just in time for the Inland Waterways Association National Trailboat Festival, which was held at Staveley over the bank holiday weekend in late May 2016.

The first boat passed through the lock on 26 May, and it was officially opened on 28 May. However, once the festival was over, a clay bund had been reinstated above the lock by 16 June, and the pound below it drained, in order to allow work to continue on the weir and spillway which carries excess water back to the River Rother. Although the track on the mineral line has been removed, its route may be reused to provide access to a maintenance depot, for the High Speed 2 (HS2) railway project. On 1 October 2017, the Trust received permission from Network Rail to investigate the bridge remains, and the infill was removed over the next few days.

Chesterfield Waterside
The southern end of the canal in Chesterfield is being redeveloped as part of a £300 million project called Chesterfield Waterside. This will provide housing and amenities in an area which is currently derelict land. The project involves the creation of a short length of new canal to create an island in the centre of the site, navigation being restored to a length of the river, and a new basin at the southern edge of the site. Funding for the canal work was provided by a grant of £500,000 from the East Midlands Development Agency. The basin was completed and was opened in October 2009, although it remains isolated from the River Rother, to which it will be connected by a lock, while a back-pumping facility will supply it with water. Outline planning permission for the whole site was granted on 15 March 2010. In April 2012, the project was awarded a £2.4 million grant by the Sheffield City Region Local Enterprise Partnership Board. The funding was for the connecting lock and the provision of sewers and access roads, to facilitate the subsequent development of the 62 acre site.

Closing the gap
With the canal open from West Stockwith to the eastern portal of Norwood Tunnel, and from Chesterfield to Staveley, there were less than 9 mi left to be restored by 2017. Whereas most of the work so far had involved restoration of the existing channel, this section includes some significant engineering challenges, with completely new construction needed to bypass the housing estate in Killamarsh, built over the original line in the 1970s, and the replacement of Norwood Tunnel with an alternative route. The Chesterfield Canal Partnership, an association of local and regional authorities, including the Chesterfield Canal Trust and other interested parties, have produced detailed plans for all of the work required to reinstate the missing section.

In 2004, the engineers Jacobs-Babtie examined six possible routes for the canal through Killamarsh. Of these, two were reasonably feasible, and were named Central Line East and Central Line West. At the time, the eastern route was felt to be the best option, and consultation with residents resulted in that route being included in the North East Derbyshire Local Plan. However, subsequent detailed design suggests that this may no longer be the best route. Redevelopment of the Tarran Bungalows site to the west of the village provided opportunities to reassess the western route, and it has gained support, as it would be more attractive for boaters, allowing them to access the village more readily. This route would also be considerably cheaper to construct that the eastern route. Both routes drop down into Nethermoor Lake, part of the Rother Valley Country Park and will ascend from the far side to regain the original course.

It will not be possible to reopen the Norwood Tunnel, as several sections have collapsed, the National Coal Board filled in some sections to stabilise the land above it, and when the M1 motorway was built, the tunnel below it was injection grouted, to ensure there would not be a subsequent collapse. The engineers Arup considered options for this section in 2007, proposing a route running largely on the surface, and crossing the site of the Kiveton Park Colliery, which closed in 1994. It involved an extra six locks at the top of the Norwood flight of 13 locks, passing under the motorway by using a farm underpass, and then descending through another six new locks, to enter the final 460 yd of the tunnel. In 2012 the UK Government's announcement of the preferred route for the HS2 rail line extension to Leeds indicated that the line would conflict with the planned restoration of the Canal in several places. The Chesterfield Canal Trust campaigned to have the needs of the Canal taken into account during the consultation process. The July 2017 announcement on the phase 2b route to be developed removed blight from the canal route. This revised route for the HS2 railway resulted in other options being considered for the tunnel replacement route. One is for a short tunnel at the western end, starting at the existing western portal, but heading in a different direction. The tunnel would be some 660 yd long, passing under the route of HS2 and the M1 motorway, allowing HS2 to be constructed without interfering with navigation. The scheme would save the need to construct six locks, and engineers were assessing the cost of such a scheme in late 2017.

In 2020, the Trust submitted a planning application covering reinstatement of the remaining 1.6 mi section that lies within the borough of Chesterfield. HS2 opposed the application, on the basis that the two projects were incompatible, but in February 2021, an agreement was reached between the Trust and HS2 to allow both to be constructed. Planning permission was then granted for the reinstatement from Staveley to Renishaw, including Brindley's Puddle Bank which carries the canal across the valley of the River Doe Lea. The canal was breached after closure to prevent flooding by the river, which passed under the canal in a culvert, and the new crossing will feature a 120 ft aqueduct. At Renishaw, 0.5 mi of canal were restored in 2010 but have since become derelict again, due to a failure to reach agreement with the Environment Agency over water supplies. That issue should be resolved as part of the development. Meanwhile, Staveley Town Basin has been renamed Staveley Waterside, as plans for a new village with a hotel and marina have been announced by Derbyshire County Council.

Features
What are often called "traditional" working narrowboats were the product of the main canal system – but the craft that plied the 46 miles between Chesterfield, Retford, and the River Trent were very different. Isolated on the fringes of the inland waterways network there were no outside canal influences to change the design of craft that were adequate for the job they had to do. The cabins were below-decks, and the boatmen always had a home ashore. Short journeys, usually with a crew of two, did not generate colourful decorations of roses-and-castles, nor did engines oust the towing horse, not even as late as the 1950s. None of the original "cuckoo" boats have survived, although the last one was extensively measured and recorded in 1976, prior to it being broken up. This has enabled the Canal Trust to create a new boat, named the Dawn Rose, which was constructed at Shireoaks Marina using traditional methods. Wood was bought in 2007, and construction started in 2011, with the boat being completed and launched in April 2015.

Norwood Tunnel was a 2884 yd, 9.25 ft and 12 ft, brick-lined canal tunnel on the line of the Chesterfield Canal with its Western Portal in Norwood, Derbyshire and its Eastern Portal in Kiveton, South Yorkshire. Although a small section of it at the eastern end may be reused as part of the restoration, most of it will remain abandoned, as it has suffered from subsidence, and parts of it have been filled in.

Drakeholes Tunnel is a second, shorter tunnel 154 yards long also without a towpath built for the Chesterfield Canal at Drakeholes, Nottinghamshire a location between the Norwood Tunnel and the River Trent. In order to allow it to be navigated by wide-beam boats, it was built 16 ft wide and 15.5 ft high. Most of it was cut through solid sandstone, and so remained unlined until 1983, when some of the rock was found to be eroding.

Hollingwood Common Tunnel is a disused navigable coalmine adit which terminated at the Chesterfield Canal near Staveley. This tunnel was 1.75 mi long, and its water level was maintained 1 ft lower than that of the canal. This level was maintained by drainage through a culvert running beneath the canal, and required the transhipment of coal at the terminus. The boats used were loaded underground within the coalmine the tunnel served; these boats were 21 ft long and 3 ft wide.

Proposals are at early stages for a link north from Killamarsh to the Sheffield and South Yorkshire Navigation along the River Rother, to be called the Rother Link. This would provide a way to reach the Chesterfield Canal without negotiating the tidal River Trent, and would create a new cruising ring, taking in the Rother Link, River Don Navigation, Stainforth and Keadby Canal, River Trent, and Chesterfield Canal.

The towpath of the canal has become a long-distance footpath and cycle route, stretching for 46 mi from Chesterfield to West Stockwith, and is signposted as the "Cuckoo Way" after the name given to the boats which worked on the canal. Parts of the towpath beside the western section of the route are also part of the Trans Pennine Trail cycle route.

The Chesterfield Canal hit international headlines in 1978. While dredging the bottom of the canal to remove rubbish a maintenance team pulled up a large chain which had a wooden plug attached to it. Later that day, it was noticed that a whirlpool had formed and it became evident that the section of the canal between Whitsunday Pie Lock and Retford Town Lock was losing water. Investigation revealed that the plug was an original engineering feature of the canal, having been installed to allow sections to be drained for future maintenance purposes. The water drained (as designed) harmlessly into the nearby river Idle. The Retford event became a national and international story at the time, and was even recorded in Lloyd's List. However, these plugs were a common feature of English canals built at that time and other instances of sudden drainage (both purposeful and accidental) have since been recorded, including emptying of the Pontcysyllte Aqueduct on the Llangollen Canal.

title: Chesterfield Canal


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District court (Norway)

In Norway, the district court (tingrett, literally "thing court") is the first court instance, and handles both criminal and civil cases. The chief judge of a district court is the chief district judge (sorenskriver).

The ruling of the district court may be appealed to the next level of court, the court of appeal, and the court of appeal's ruling may in turn be appealed to the supreme court, but only in select cases of precedential relevance admitted by the supreme court. Thus the vast majority of court cases are decided on by the district court or the court of appeal.

The term tingrett for the district courts was introduced in 2002, replacing the previous terms city court (byrett) and district court (herredsrett). Historically the district courts outside the cities were known as sorenskriveri (office of a sorenskriver), and consisted of a single (chief) district judge and often one or two assistant judges. The district usually included multiple parishes/municipalities, and the district judge was typically the highest state official within his jurisdiction, with an almost all-encompassing area of jurisdiction and an important role in the regional state administration.

Criminal cases
Since August 1995, all criminal cases which go to court are tried at the district court first. Prior to this, the most serious cases were tried in the Court of appeal (lagmannsrett). Regular trials are usually held before a panel of three judges, one professional and two lay judges. In particularly difficult cases the bench may consist of five judges (two professional and three lay judges). Verdicts and sentences are determined by a majority vote, with the lay judges' votes counting the same as the professional judge. If there is a full confession, and the defendant agrees to it, the trial may be held before a single professional judge who hears the case and determines sentencing.

Civil cases
The district court is the second tier in most civil cases, the first tier being the conciliation board (forliksråd). Certain cases cannot be held at the conciliation board, including family law, patent and trademark issues, cases against the authorities, and cases where an independent complaints committee has issued an opinion. In addition, larger cases where both sides are representad by attorneys, and other cases where extrajudicial mediation have taken place, may bypass the conciliation board and go straight to the district court. The district court also hears appeals from the conciliation board.

Civil cases are normally held before a single professional judge, but either side can demand two lay judges be seated as well. In some cases the law demands that there be two lay judges with competence in the subject matter.

In 2021, there are 23 district courts in Norway.

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title: District court (Norway)


text:
Fokker–Planck equation

In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. The Fokker-Planck equation has multiple applications in information theory, graph theory, data science, finance, economics etc.

It is named after Adriaan Fokker and Max Planck, who described it in 1914 and 1917. It is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered it in 1931. When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski), and in this context it is equivalent to the convection–diffusion equation. When applied to particle position and momentum distributions, it is known as the Klein–Kramers equation. The case with zero diffusion is the continuity equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion.

The first consistent microscopic derivation of the Fokker–Planck equation in the single scheme of classical and quantum mechanics was performed by Nikolay Bogoliubov and Nikolay Krylov.

One dimension
In one spatial dimension x, for an Itô process driven by the standard Wiener process $$W_t$$ and described by the stochastic differential equation (SDE) $$dX_t = \mu(X_t, t) \,dt + \sigma(X_t, t) \,dW_t$$ with drift $$\mu(X_t, t)$$ and diffusion coefficient $$D(X_t, t) = \sigma^2(X_t, t)/2$$, the Fokker–Planck equation for the probability density $$p(x, t)$$ of the random variable $$X_t$$ is $$ In the following, use $$\sigma = \sqrt{2D}$$.

Define the infinitesimal generator $$\mathcal{L}$$ (the following can be found in Ref. ): $$ \mathcal{L}p(X_t) = \lim_{\Delta t \to 0} \frac1{\Delta t}\left(\mathbb{E}\big[p(X_{t + \Delta t}) \mid X_t = x \big] - p(x)\right). $$

The transition probability $$\mathbb{P}_{t, t'}(x \mid x')$$, the probability of going from $$(t', x')$$ to $$(t, x)$$, is introduced here; the expectation can be written as $$ \mathbb{E}(p(X_{t + \Delta t}) \mid X_t = x) = \int p(y) \, \mathbb{P}_{t + \Delta t,t}(y \mid x) \,dy. $$ Now we replace in the definition of $$\mathcal{L}$$, multiply by $$\mathbb{P}_{t, t'}(x \mid x')$$ and integrate over $$dx$$. The limit is taken on $$ \int p(y) \int \mathbb{P}_{t + \Delta t, t}(y \mid x)\,\mathbb{P}_{t, t'}(x \mid x') \,dx \,dy - \int p(x) \, \mathbb{P}_{t, t'}(x \mid x') \,dx. $$ Note now that $$ \int \mathbb{P}_{t + \Delta t, t}(y \mid x) \, \mathbb{P}_{t, t'}(x \mid x') \,dx = \mathbb{P}_{t + \Delta t, t'}(y \mid x'), $$ which is the Chapman–Kolmogorov theorem. Changing the dummy variable $$y$$ to $$x$$, one gets $$ \begin{align} \int p(x) \lim_{\Delta t \to 0} \frac1{\Delta t} \left( \mathbb{P}_{t + \Delta t, t'}(x \mid x') - \mathbb{P}_{t, t'}(x \mid x') \right) \,dx, \end{align} $$ which is a time derivative. Finally we arrive to $$ \int [\mathcal{L}p(x)] \mathbb{P}_{t, t'}(x \mid x') \,dx = \int p(x) \, \partial_t \mathbb{P}_{t, t'}(x \mid x') \,dx. $$ From here, the Kolmogorov backward equation can be deduced. If we instead use the adjoint operator of $$\mathcal{L}$$, $$\mathcal{L}^\dagger$$, defined such that $$ \int [\mathcal{L}p(x)] \mathbb{P}_{t, t'}(x \mid x') \,dx = \int p(x) [\mathcal{L}^\dagger \mathbb{P}_{t, t'}(x \mid x')] \,dx, $$ then we arrive to the Kolmogorov forward equation, or Fokker–Planck equation, which, simplifying the notation $$p(x, t) = \mathbb{P}_{t, t'}(x \mid x')$$, in its differential form reads $$ \mathcal{L}^\dagger p(x, t) = \partial_t p(x, t). $$

Remains the issue of defining explicitly $$\mathcal{L}$$. This can be done taking the expectation from the integral form of the Itô's lemma: $$ \mathbb{E}\big(p(X_t)\big) = p(X_0) + \mathbb{E}\left(\int_0^t \left(\partial_t + \mu\partial_x + \frac{\sigma^2}{2}\partial_x^2 \right) p(X_{t'}) \,dt'\right). $$

The part that depends on $$dW_t$$ vanished because of the martingale property.

Then, for a particle subject to an Itô equation, using $$ \mathcal{L} = \mu\partial_x + \frac{\sigma^2}{2}\partial_x^2, $$ it can be easily calculated, using integration by parts, that $$ \mathcal{L}^\dagger = -\partial_x(\mu \cdot) + \frac12 \partial_x^2(\sigma^2 \cdot), $$ which bring us to the Fokker–Planck equation: $$ \partial_t p(x, t) = -\partial_x \big(\mu(x, t) \cdot p(x, t)\big) + \partial_x^2\left(\frac{\sigma(x, t)^2}{2} \, p(x,t)\right). $$

While the Fokker–Planck equation is used with problems where the initial distribution is known, if the problem is to know the distribution at previous times, the Feynman–Kac formula can be used, which is a consequence of the Kolmogorov backward equation.

The stochastic process defined above in the Itô sense can be rewritten within the Stratonovich convention as a Stratonovich SDE: $$dX_t = \left[\mu(X_t, t) - \frac{1}{2} \frac{\partial}{\partial X_t}D(X_t, t)\right] \,dt + \sqrt{2 D(X_t, t)} \circ dW_t.$$ It includes an added noise-induced drift term due to diffusion gradient effects if the noise is state-dependent. This convention is more often used in physical applications. Indeed, it is well known that any solution to the Stratonovich SDE is a solution to the Itô SDE.

The zero-drift equation with constant diffusion can be considered as a model of classical Brownian motion: $$\frac{\partial}{\partial t} p(x, t) = D_0\frac{\partial^2}{\partial x^2}\left[p(x, t)\right].$$

This model has discrete spectrum of solutions if the condition of fixed boundaries is added for $$\{0 \leq x \leq L\}$$: $$\begin{align} p(0, t) &= p(L, t) = 0, \\ p(x, 0) &= p_0(x). \end{align}$$

It has been shown that in this case an analytical spectrum of solutions allows deriving a local uncertainty relation for the coordinate-velocity phase volume: $$ \Delta x \, \Delta v \geq D_0. $$ Here $$D_0$$ is a minimal value of a corresponding diffusion spectrum $$D_j$$, while $$\Delta x$$ and $$\Delta v$$ represent the uncertainty of coordinate–velocity definition.

Higher dimensions
More generally, if

$$d\mathbf{X}_t = \boldsymbol{\mu}(\mathbf{X}_t,t)\,dt + \boldsymbol{\sigma}(\mathbf{X}_t,t)\,d\mathbf{W}_t,$$

where $$\mathbf{X}_t$$ and $$\boldsymbol{\mu}(\mathbf{X}_t,t)$$ are $N$-dimensional vectors, $$\boldsymbol{\sigma}(\mathbf{X}_t,t)$$ is an $$N \times M$$ matrix and $$\mathbf{W}_t$$ is an M-dimensional standard Wiener process, the probability density $$p(\mathbf{x},t)$$ for $$\mathbf{X}_t$$ satisfies the Fokker–Planck equation$$with drift vector $$\boldsymbol{\mu} = (\mu_1,\ldots,\mu_N)$$ and diffusion tensor $\mathbf{D} = \frac{1}{2} \boldsymbol{\sigma\sigma}^\mathsf{T}$, i.e.$$D_{ij}(\mathbf{x},t) = \frac{1}{2}\sum_{k=1}^M \sigma_{ik}(\mathbf{x},t) \sigma_{jk}(\mathbf{x},t).$$

If instead of an Itô SDE, a Stratonovich SDE is considered,

$$d\mathbf{X}_t = \boldsymbol{\mu}(\mathbf{X}_t,t)\,dt + \boldsymbol{\sigma}(\mathbf{X}_t,t)\circ d\mathbf{W}_t,$$

the Fokker–Planck equation will read:

$$\frac{\partial p(\mathbf{x},t)}{\partial t} = -\sum_{i=1}^N \frac{\partial}{\partial x_i} \left[ \mu_i(\mathbf{x},t) \, p(\mathbf{x},t) \right] + \frac{1}{2} \sum_{k=1}^M \sum_{i=1}^{N} \frac{\partial}{\partial x_i} \left\{ \sigma_{ik}(\mathbf{x},t) \sum_{j=1}^{N} \frac{\partial}{\partial x_j} \left[  \sigma_{jk}(\mathbf{x},t) \, p(\mathbf{x},t) \right] \right\}$$

Generalization
In general, the Fokker–Planck equations are a special case to the general Kolmogorov forward equation

$$\partial_t \rho = \mathcal{A}^*\rho$$

where the linear operator $$\mathcal{A}^*$$ is the Hermitian adjoint to the infinitesimal generator for the Markov process.

Wiener process
A standard scalar Wiener process is generated by the stochastic differential equation

$$dX_t = dW_t.$$

Here the drift term is zero and the diffusion coefficient is 1/2. Thus the corresponding Fokker–Planck equation is

$$ \frac{\partial p(x,t)}{\partial t} = \frac{1}{2} \frac{\partial^2 p(x,t)}{\partial x^2}, $$

which is the simplest form of a diffusion equation. If the initial condition is $$p(x,0) = \delta(x)$$, the solution is

$$ p(x,t) = \frac{1}{\sqrt{2 \pi t}}e^{-{x^2}/({2t})}.$$

Boltzmann distribution at the thermodynamic equilibrium
The overdamped Langevin equation$$dx_t = -\frac{1}{k_\text{B}T} (\nabla_x U)dt + dW_t$$gives $\partial_t p = \frac 1 2 \nabla\cdot \left(\frac{p}{k_\text{B}T} \nabla U + \nabla p\right)$. The Boltzmann distribution $$p(x) \propto e^{- U(x)/k_\text{B} T}$$is an equilibrium distribution, and assuming $$U$$ grows sufficiently rapidly (that is, the potential well is deep enough to confine the particle), the Boltzmann distribution is the unique equilibrium.

Ornstein–Uhlenbeck process
The Ornstein–Uhlenbeck process is a process defined as

$$dX_t = -a X_t \, dt + \sigma \, dW_t.$$

with $$a>0$$. Physically, this equation can be motivated as follows: a particle of mass $$ m $$ with velocity $$ V_t$$ moving in a medium, e.g., a fluid, will experience a friction force which resists motion whose magnitude can be approximated as being proportional to particle's velocity $$ -a V_t$$ with $$ a = \mathrm{constant} $$. Other particles in the medium will randomly kick the particle as they collide with it and this effect can be approximated by a white noise term; $$ \sigma (d W_t/dt) $$. Newton's second law is written as

$$ m \frac{dV_t}{dt}=-a V_t +\sigma \frac{dW_t}{dt}. $$

Taking $$ m = 1$$ for simplicity and changing the notation as $$ V_t\rightarrow X_t$$ leads to the familiar form $$dX_t = -a X_t dt + \sigma dW_t$$.

The corresponding Fokker–Planck equation is $$ \frac{\partial p(x,t)}{\partial t} = a \frac{\partial}{\partial x}\left(x \,p(x,t)\right) + \frac{\sigma^2}{2} \frac{\partial^2 p(x,t)}{\partial x^2}, $$

The stationary solution ($$\partial_t p = 0$$) is $$p_{ss}(x) = \sqrt{\frac{a}{\pi \sigma^2}} e^{-{ax^2}/{\sigma^2}}.$$

Plasma physics
In plasma physics, the distribution function for a particle species $$s$$, $$p_s (\mathbf{x},\mathbf{v},t)$$, takes the place of the probability density function. The corresponding Boltzmann equation is given by

$$\frac{\partial p_s}{\partial t} + \mathbf{v} \cdot \boldsymbol{\nabla} p_s + \frac{Z_s e}{m_s} \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right) \cdot \boldsymbol{\nabla}_v p_s = -\frac{\partial}{\partial v_i} \left(p_s \langle\Delta v_i\rangle\right) + \frac{1}{2} \frac{\partial^2}{\partial v_i \, \partial v_j} \left(p_s \langle\Delta v_i \, \Delta v_j\rangle\right),$$

where the third term includes the particle acceleration due to the Lorentz force and the Fokker–Planck term at the right-hand side represents the effects of particle collisions. The quantities $$\langle\Delta v_i\rangle$$ and $$\langle\Delta v_i \, \Delta v_j\rangle$$ are the average change in velocity a particle of type $$s$$ experiences due to collisions with all other particle species in unit time. Expressions for these quantities are given elsewhere. If collisions are ignored, the Boltzmann equation reduces to the Vlasov equation.

Smoluchowski diffusion equation
Consider an overdamped Brownian particle under external force $$F(r)$$: $$m\ddot{r} = - \gamma \dot{r} + F(r) + \sigma \xi(t)$$where the $$m\ddot r$$ term is negligible (the meaning of "overdamped"). Thus, it is just $$\gamma \, dr = F(r)\, dt + \sigma \, dW_t$$. The Fokker–Planck equation for this particle is the Smoluchowski diffusion equation: $$\partial_t P(r,t| r_0, t_0) = \nabla \cdot [D (\nabla - \beta F(r)) P(r,t| r_0, t_0)] $$Where $$D$$ is the diffusion constant and $$\beta = \frac{1}{k_\text{B} T}$$. The importance of this equation is it allows for both the inclusion of the effect of temperature on the system of particles and a spatially dependent diffusion constant.

Starting with the Langevin Equation of a Brownian particle in external field $$F(r)$$, where $$\gamma$$ is the friction term, $$\xi$$ is a fluctuating force on the particle, and $$\sigma$$ is the amplitude of the fluctuation.

$$m\ddot{r} = - \gamma \dot{r} + F(r) + \sigma \xi(t)$$

At equilibrium the frictional force is much greater than the inertial force, $$\left\vert \gamma \dot{r} \right\vert \gg \left\vert m \ddot{r} \right\vert$$. Therefore, the Langevin equation becomes,

$$\gamma \dot{r} = F(r) + \sigma \xi(t)$$

Which generates the following Fokker–Planck equation,

$$\partial_t P(r,t|r_0,t_0) = \left(\nabla^2\frac{\sigma^2}{2 \gamma^2} - \nabla \cdot \frac{F(r)}{\gamma}\right) P(r,t|r_0,t_0) $$

Rearranging the Fokker–Planck equation,

$$\partial_t P(r,t|r_0,t_0)= \nabla \cdot \left( \nabla D- \frac{F(r)}{\gamma}\right) P(r,t|r_0,t_0)$$

Where $$D = \frac{\sigma^2}{2 \gamma^2}$$. Note, the diffusion coefficient may not necessarily be spatially independent if $$\sigma$$ or $$\gamma$$ are spatially dependent.

Next, the total number of particles in any particular volume is given by,

$$N_V (t| r_0, t_0) = \int_V dr P(r,t|r_0,t_0)$$

Therefore, the flux of particles can be determined by taking the time derivative of the number of particles in a given volume, plugging in the Fokker–Planck equation, and then applying Gauss's Theorem.

$$\partial_t N_V (t|r_0, t_0) = \int_V dV \nabla \cdot\left( \nabla D- \frac{F(r)}{\gamma}\right) P(r,t|r_0, t_0) = \int_{\partial V} d\mathbf{a} \cdot j(r,t|r_0, t_0)$$

$$j(r,t|r_0, t_0) = \left( \nabla D- \frac{F(r)}{\gamma}\right)P(r,t|r_0, t_0)$$

In equilibrium, it is assumed that the flux goes to zero. Therefore, Boltzmann statistics can be applied for the probability of a particles location at equilibrium, where $$F(r) = -\nabla U(r)$$ is a conservative force and the probability of a particle being in a state $$r$$ is given as $$P(r,t|r_0, t_0) = \frac{e^{-\beta U(r)}}{Z}$$.

$$j(r,t|r_0, t_0) = \left( \nabla D- \frac{F(r)}{\gamma}\right)\frac{e^{-\beta U(r)}}{Z} = 0$$

$$\Rightarrow \nabla D   = F(r) \left(\frac{1}{\gamma} - D \beta\right)$$

This relation is a realization of the fluctuation–dissipation theorem. Now applying $$ \nabla \cdot \nabla $$ to $$D P(r,t|r_0, t_0)$$ and using the Fluctuation-dissipation theorem,

$$\begin{align} \nabla \cdot \nabla D P(r,t|r_0,t_0) &= \nabla \cdot D \nabla P(r,t|r_0,t_0)+ \nabla \cdot P(r,t|r_0,t_0) \nabla D \\ &=\nabla \cdot D \nabla P(r,t|r_0,t_0)+\nabla \cdot P(r,t|r_0,t_0) \frac{F(r)}{\gamma} - \nabla \cdot P(r,t|r_0,t_0) D \beta F(r) \end{align}$$

Rearranging,

$$ \Rightarrow \nabla \cdot \left( \nabla D- \frac{F(r)}{\gamma}\right)P(r,t|r_0,t_0)= \nabla \cdot D(\nabla-\beta F(r)) P(r,t|r_0,t_0)$$

Therefore, the Fokker–Planck equation becomes the Smoluchowski equation, $$\partial_t P(r,t| r_0, t_0) = \nabla \cdot D (\nabla - \beta F(r)) P(r,t| r_0, t_0) $$ for an arbitrary force $$F(r)$$.

Computational considerations
Brownian motion follows the Langevin equation, which can be solved for many different stochastic forcings with results being averaged (canonical ensemble in molecular dynamics). However, instead of this computationally intensive approach, one can use the Fokker–Planck equation and consider the probability $$p(\mathbf{v}, t)\,d\mathbf{v}$$ of the particle having a velocity in the interval $$(\mathbf{v}, \mathbf{v} + d\mathbf{v})$$ when it starts its motion with $$\mathbf{v}_0$$ at time 0.

1-D linear potential example
Brownian dynamics in one dimension is simple.

Theory
Starting with a linear potential of the form $$U(x) = cx$$ the corresponding Smoluchowski equation becomes,

$$\partial_t P(x,t| x_0, t_0) = \partial_x D (\partial_x + \beta c) P(x,t| x_0, t_0) $$

Where the diffusion constant, $$D$$, is constant over space and time. The boundary conditions are such that the probability vanishes at $$x \rightarrow \pm \infin $$ with an initial condition of the ensemble of particles starting in the same place, $$P(x,t|x_0,t_0)= \delta (x-x_0) $$.

Defining $$\tau = D t $$ and $$b = \beta c $$ and applying the coordinate transformation,

$$y = x +\tau b ,\ \ \ y_0= x_0 + \tau_0 b $$

With $$P(x, t, |x_0, t_0) = q(y, \tau|y_0, \tau_0)$$ the Smoluchowki equation becomes, $$\partial_\tau q(y, \tau| y_0, \tau_0) =\partial_y^2 q(y, \tau| y_0, \tau_0)$$

Which is the free diffusion equation with solution, $$q(y, \tau| y_0, \tau_0)= \frac{1}{\sqrt {4 \pi (\tau - \tau_0)}} e^{ -\frac{(y-y_0)^2}{4(\tau-\tau_0)} }$$

And after transforming back to the original coordinates, $$P(x, t | x_0, t_0)= \frac{1}{\sqrt{4 \pi D (t - t_0)}} \exp {\left[{ -\frac{(x-x_0+ D \beta c(t-t_0))^2}{4D(t-t_0)}} \right]}$$

Simulation
The simulation on the right was completed using a Brownian dynamics simulation. Starting with a Langevin equation for the system, $$m\ddot{x} = - \gamma \dot{x} -c + \sigma \xi(t)$$ where $$\gamma$$ is the friction term, $$\xi$$ is a fluctuating force on the particle, and $$\sigma$$ is the amplitude of the fluctuation. At equilibrium the frictional force is much greater than the inertial force, $$\left| \gamma \dot{x} \right| \gg \left| m \ddot{x} \right|$$. Therefore, the Langevin equation becomes, $$\gamma \dot{x} = -c + \sigma \xi(t)$$

For the Brownian dynamic simulation the fluctuation force $$\xi(t)$$ is assumed to be Gaussian with the amplitude being dependent of the temperature of the system $\sigma = \sqrt{2\gamma k_\text{B} T}$. Rewriting the Langevin equation,

$$\frac{dx}{dt}=-D \beta c + \sqrt{2D}\xi(t)$$ where $D = \frac{k_\text{B}T}{\gamma}$ is the Einstein relation. The integration of this equation was done using the Euler–Maruyama method to numerically approximate the path of this Brownian particle.

Solution
Being a partial differential equation, the Fokker–Planck equation can be solved analytically only in special cases. A formal analogy of the Fokker–Planck equation with the Schrödinger equation allows the use of advanced operator techniques known from quantum mechanics for its solution in a number of cases. Furthermore, in the case of overdamped dynamics when the Fokker–Planck equation contains second partial derivatives with respect to all spatial variables, the equation can be written in the form of a master equation that can easily be solved numerically. In many applications, one is only interested in the steady-state probability distribution $$ p_0(x)$$, which can be found from $\frac{\partial p(x,t)}{\partial t} = 0$. The computation of mean first passage times and splitting probabilities can be reduced to the solution of an ordinary differential equation which is intimately related to the Fokker–Planck equation.

Particular cases with known solution and inversion
In mathematical finance for volatility smile modeling of options via local volatility, one has the problem of deriving a diffusion coefficient $${\sigma}(\mathbf{X}_t,t)$$ consistent with a probability density obtained from market option quotes. The problem is therefore an inversion of the Fokker–Planck equation: Given the density f(x,t) of the option underlying X deduced from the option market, one aims at finding the local volatility $${\sigma}(\mathbf{X}_t,t)$$ consistent with f. This is an inverse problem that has been solved in general by Dupire (1994, 1997) with a non-parametric solution. Brigo and Mercurio (2002, 2003) propose a solution in parametric form via a particular local volatility $${\sigma}(\mathbf{X}_t,t)$$ consistent with a solution of the Fokker–Planck equation given by a mixture model. More information is available also in Fengler (2008), Gatheral (2008), and Musiela and Rutkowski (2008).

Fokker–Planck equation and path integral
Every Fokker–Planck equation is equivalent to a path integral. The path integral formulation is an excellent starting point for the application of field theory methods. This is used, for instance, in critical dynamics.

A derivation of the path integral is possible in a similar way as in quantum mechanics. The derivation for a Fokker–Planck equation with one variable $$x$$ is as follows. Start by inserting a delta function and then integrating by parts:

$$\begin{align} \frac{\partial }{\partial t} p{\left( x', t\right)} & = - \frac{\partial }{\partial x'} \left[ D_1(x',t) p(x',t) \right] + \frac{\partial^2 }{\partial {x'}^2} \left[ D_2(x',t) p(x',t) \right] \\[1ex] & = \int_{-\infty}^{\infty} dx\left[ \left( D_{1}{\left( x,t\right)} \frac{\partial }{\partial x} + D_2{ \left( x,t\right)} \frac{\partial^2}{\partial x^2}\right) \delta{\left( x' -x\right)} \right] p(x,t). \end{align}$$

The $$x$$-derivatives here only act on the $$\delta$$-function, not on $$p(x,t)$$. Integrate over a time interval $$\varepsilon$$,

$$p(x', t + \varepsilon) =\int_{-\infty}^\infty \, \mathrm{d}x\left(\left( 1+\varepsilon \left[ D_1(x,t) \frac \partial {\partial x} + D_2(x,t) \frac{\partial^2}{\partial x^2}\right]\right) \delta(x' - x) \right) p(x,t)+O(\varepsilon^2).$$

Insert the Fourier integral

$$\delta{\left( x' - x\right)} = \int_{-i\infty}^{i\infty} \frac{\mathrm{d} \tilde{x}}{2\pi i} e^{\tilde{x} {\left( x - x'\right)}}$$

for the $$\delta$$-function,

$$\begin{align} p(x', t+\varepsilon) & = \int_{-\infty}^\infty \mathrm{d}x \int_{-i\infty}^{i\infty} \frac{\mathrm{d}\tilde{x}}{2\pi i} \left(1+\varepsilon \left[ \tilde{x} D_1(x,t) +\tilde{x}^2 D_2(x,t) \right] \right) e^{\tilde{x} (x - x')}p(x,t) +O(\varepsilon^2) \\[5pt] & =\int_{-\infty}^\infty \mathrm{d}x \int_{-i\infty}^{i\infty} \frac{\mathrm{d}\tilde{x}}{2\pi i}\exp \left( \varepsilon \left[ -\tilde{x}\frac{(x'- x) }\varepsilon + \tilde{x} D_1(x,t) +\tilde{x}^2 D_2(x,t) \right] \right) p(x,t) +O(\varepsilon^2). \end{align}$$

This equation expresses $$p(x', t+\varepsilon)$$ as functional of $$p(x,t)$$. Iterating $$(t'-t)/\varepsilon$$ times and performing the limit $$\varepsilon \rightarrow 0$$ gives a path integral with action

$$S=\int \mathrm{d}t\left[ \tilde{x} D_1 (x,t) +\tilde{x}^2 D_2 (x,t) -\tilde{x}\frac{\partial x}{\partial t} \right].$$

The variables $$\tilde{x}$$ conjugate to $$x$$ are called "response variables".

Although formally equivalent, different problems may be solved more easily in the Fokker–Planck equation or the path integral formulation. The equilibrium distribution for instance may be obtained more directly from the Fokker–Planck equation.

title: Fokker–Planck equation


text:
Knot invariant

In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic ), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, "a knot invariant is a rule that assigns to any knot $K$ a quantity $φ(K)$ such that if $K$ and $K'$ are equivalent then $φ(K) = φ(K')$." ). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification, both in "enumeration" and "duplication removal".

"A knot invariant is a quantity defined on the set of all knots, which takes the same value for any two equivalent knots. For example, a knot group is a knot invariant."

"Typically a knot invariant is a combinatorial quantity defined on knot diagrams. Thus if two knot diagrams differ with respect to some knot invariant, they must represent different knots. However, as is generally the case with topological invariants, if two knot diagrams share the same values with respect to a [single] knot invariant, then we still cannot conclude that the knots are the same."

From the modern perspective, it is natural to define a knot invariant from a knot diagram. Of course, it must be unchanged (that is to say, invariant) under the Reidemeister moves ("triangular moves" ). Tricolorability (and n-colorability) is a particularly simple and common example. Other examples are knot polynomials, such as the Jones polynomial, which are currently among the most useful invariants for distinguishing knots from one another, though currently it is not known whether there exists a knot polynomial which distinguishes all knots from each other. However, there are invariants which distinguish the unknot from all other knots, such as Khovanov homology and knot Floer homology.

Other invariants can be defined by considering some integer-valued function of knot diagrams and taking its minimum value over all possible diagrams of a given knot. This category includes the crossing number, which is the minimum number of crossings for any diagram of the knot, and the bridge number, which is the minimum number of bridges for any diagram of the knot.

Historically, many of the early knot invariants are not defined by first selecting a diagram but defined intrinsically, which can make computing some of these invariants a challenge. For example, knot genus is particularly tricky to compute, but can be effective (for instance, in distinguishing mutants).

The complement of a knot itself (as a topological space) is known to be a "complete invariant" of the knot by the Gordon–Luecke theorem in the sense that it distinguishes the given knot from all other knots up to ambient isotopy and mirror image. Some invariants associated with the knot complement include the knot group which is just the fundamental group of the complement. The knot quandle is also a complete invariant in this sense but it is difficult to determine if two quandles are isomorphic. The peripheral subgroup can also work as a complete invariant.

By Mostow–Prasad rigidity, the hyperbolic structure on the complement of a hyperbolic link is unique, which means the hyperbolic volume is an invariant for these knots and links. Volume, and other hyperbolic invariants, have proven very effective, utilized in some of the extensive efforts at knot tabulation.

In recent years, there has been much interest in homological invariants of knots which categorify well-known invariants. Heegaard Floer homology is a homology theory whose Euler characteristic is the Alexander polynomial of the knot. It has been proven effective in deducing new results about the classical invariants. Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial. This has recently been shown to be useful in obtaining bounds on slice genus whose earlier proofs required gauge theory. Mikhail Khovanov and Lev Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants. Catharina Stroppel gave a representation theoretic interpretation of Khovanov homology by categorifying quantum group invariants.

There is also growing interest from both knot theorists and scientists in understanding "physical" or geometric properties of knots and relating it to topological invariants and knot type. An old result in this direction is the Fáry–Milnor theorem states that if the total curvature of a knot $inv(L) = inv(L')$ in $$\R^3$$ satisfies

 * $$\oint_K \kappa \,ds \leq 4\pi,$$

where $L$ is the curvature at $L'$, then $K$ is an unknot. Therefore, for knotted curves,

 * $$\oint_K \kappa\,ds > 4\pi.\,$$

An example of a "physical" invariant is ropelength, which is the length of unit-diameter rope needed to realize a particular knot type.

Other invariants

 * (or Vassiliev or Vassiliev–Goussarov invariant)
 * (or Vassiliev or Vassiliev–Goussarov invariant)

title: Knot invariant


text:
Eugene McCarthy

Eugene Joseph McCarthy (March 29, 1916 – December 10, 2005) was an American politician, writer, and academic from Minnesota. He served in the United States House of Representatives from 1949 to 1959 and the United States Senate from 1959 to 1971. McCarthy sought the Democratic presidential nomination in the 1968 election, challenging incumbent Lyndon B. Johnson on an anti-Vietnam War platform. McCarthy unsuccessfully ran for U.S. president four more times.

Born in Watkins, Minnesota, McCarthy became an economics professor after earning a graduate degree from the University of Minnesota. He served as a code breaker for the United States Department of War during World War II. McCarthy became a member of the Minnesota Democratic–Farmer–Labor Party (the state affiliate of the Democratic Party) and in 1948 was elected to the House of Representatives, where he served until being elected to the U.S. Senate in 1958. McCarthy was a prominent supporter of Adlai Stevenson II for the Democratic presidential nomination in 1960, and was himself a candidate for the Democratic vice-presidential nomination in 1964. He co-sponsored the Immigration and Nationality Act of 1965, though he later expressed regret about its impact and became a member of the Federation for American Immigration Reform.

As the 1960s progressed, McCarthy emerged as a prominent opponent of Johnson's handling of the Vietnam War. After Robert F. Kennedy declined the request of a group of antiwar Democrats to challenge Johnson in the 1968 Democratic primaries, McCarthy entered the race on an antiwar platform. Though he was initially given little chance of winning, the Tet Offensive galvanized opposition to the war, and McCarthy finished in a strong second place in the New Hampshire primary. After that, Kennedy entered the race, and Johnson announced that he would not seek reelection. McCarthy and Kennedy each won several primaries before Kennedy was assassinated in June 1968. McCarthy won the plurality of the primary vote, which at the time was non-binding, losing at the 1968 Democratic National Convention to Vice President Hubert Humphrey, Johnson's preferred candidate.

McCarthy did not seek reelection in the 1970 Senate election. He sought the Democratic presidential nomination in 1972 but fared poorly in the primaries. He ran in several more races after that but was never elected to another office. He ran as an Independent in the 1976 presidential election and won 0.9% of the popular vote. He was a plaintiff in the landmark campaign finance case Buckley v. Valeo and supported Ronald Reagan in the 1980 presidential election.

Early life
McCarthy was born in Watkins, Minnesota. He was the son of a deeply religious Catholic woman of German ancestry, Anna Baden McCarthy, and a strong-willed man of Irish descent, Michael John McCarthy Jr., a postmaster and cattle buyer.

McCarthy grew up in Watkins with his parents and three siblings. He attended St. Anthony's Catholic School in Watkins, and spent hours reading his aunt's Harvard Classics. He was influenced by the monks at nearby St. John's Abbey and University in Collegeville, Minnesota, and attended school there, at Saint John's Preparatory School, from which he graduated in 1932. He also went to college at Saint John's University, graduating in 1935. McCarthy earned his master's degree from the University of Minnesota in 1939. He taught in public schools in Minnesota and North Dakota from 1935 to 1940, when he became a professor of economics and education at St. John's, working there from 1940 to 1943.

While at St. John's, he coached the hockey team for one season.

In 1943, considering the contemplative life of a monk, he became a Benedictine novice at Saint John's Abbey. After nine months as a monk he left the monastery, causing a fellow novice to say, "It was like losing a 20-game winner". He enlisted in the Army, serving as a code breaker for the Military Intelligence Division of the War Department in Washington, D.C. in 1944. He was then an instructor in sociology and economics at the College of St. Thomas in St. Paul, Minnesota, from 1946 to 1949.

U.S. House of Representatives
McCarthy became a member of the Minnesota Democratic-Farmer-Labor Party. In 1948 he was elected to the United States House of Representatives with labor and Catholic support, representing Minnesota's 4th congressional district until 1959. He became the leader of young liberals, predominately from the Midwest, called "McCarthy's Marauders".

In 1952 he engaged Wisconsin Senator Joseph McCarthy (no relation) in a nationally televised debate in which he parodied the Senator's arguments to "prove" that General Douglas MacArthur had been a communist pawn. In 1958 he was elected to the U.S. Senate.

U.S. Senate
He served as a member of (among other committees) the powerful Senate Foreign Relations Committee. McCarthy became known to a larger audience in 1960 when he supported twice-defeated presidential candidate Adlai Stevenson for the Democratic nomination. He pleaded during his speech nominating Stevenson, "Do not reject this man who made us all proud to be called Democrats!" He joked about his own merits as a candidate, "I'm twice as liberal as Hubert Humphrey, twice as intelligent as Stuart Symington, and twice as Catholic as Jack Kennedy." He was considered as a possible running mate for Lyndon Johnson in 1964, only to see fellow Minnesota Senator Humphrey chosen for that position.

McCarthy voted in favor of the Civil Rights Act of 1960, the 24th Amendment to the U.S. Constitution, the Civil Rights Act of 1964, the Voting Rights Act of 1965, and the Medicare program. He did not vote on the Civil Rights Act of 1968 or on the confirmation of Thurgood Marshall to the U.S. Supreme Court.

Along with Ted Kennedy, McCarthy was one of the original co-sponsors of the Immigration Act of 1965. He later regretted this, noting that "unrecognized by virtually all of the bill's supporters were provisions which would eventually lead to unprecedented growth in numbers and the transfer of policy control from the elected representatives of the American people to individuals wishing to bring relatives to this country". He became a member of the Federation for American Immigration Reform's board of advisors.

McCarthy met with Marxist-Leninist revolutionary Che Guevara in New York City in 1964 to discuss repairing relations between the US and Cuba. They met in journalist Lisa Howard's apartment on Park Avenue in Manhattan. The 2008 film Che: Part One depicts this event.

McCarthy challenges Johnson
In 1968, Allard K. Lowenstein and his anti-Vietnam War Dump Johnson movement recruited McCarthy to run against incumbent President Lyndon B. Johnson. Reportedly, Lowenstein first attempted to recruit Senator Robert F. Kennedy, who declined to run, then Senator George McGovern, who also declined (Kennedy eventually decided to run after the primary on March 16, 1968, and McGovern also later briefly entered the race). McCarthy entered and almost defeated Johnson in the New Hampshire Democratic primary, with the intention of influencing the federal government—then controlled by Democrats—to curtail its involvement in the Vietnam War. A number of antiwar college students and other activists from around the country traveled to New Hampshire to support McCarthy's campaign. Some antiwar students who had the long-haired, counterculture appearance of hippies chose to cut their long hair and shave off their beards in order to campaign for McCarthy door-to-door, a phenomenon that led to the informal slogan "Get clean for Gene".

McCarthy's decision to run arose partly as an outcome of Oregon Senator Wayne Morse's opposition to the war. Morse was one of two senators to vote against the Gulf of Tonkin Resolution of August 1964. He gave speeches denouncing the war before it had entered most Americans' awareness. Thereafter, several politically active Oregon Democrats asked Kennedy to run as an antiwar candidate. McCarthy also encouraged Kennedy to run. After Kennedy refused, the group asked McCarthy to run, and he responded favorably. After Kennedy entered the race and Johnson withdrew, however, McCarthy shifted his focus toward Kennedy.

McCarthy declared his candidacy on November 30, 1967, saying, "I am concerned that the Administration seems to have set no limit to the price it is willing to pay for a military victory." Political experts and the news media dismissed his candidacy, and he was given little chance of making any impact against Johnson in the primaries. But public perception of him changed following the Tet Offensive (January 30 – February 23, 1968), the aftermath of which saw many Democrats grow disillusioned with the war, and quite a few interested in an alternative to Johnson. McCarthy said, "My decision to challenge the President's position and the administration's position has been strengthened by recent announcements out of the administration. The evident intention to escalate and to intensify the war in Vietnam, and on the other hand, the absence of any positive indication or suggestion for a compromise or for a negotiated political settlement."

On December 3, 1967, McCarthy addressed the Conference of Concerned Democrats in Chicago, accusing the Johnson administration of ignoring and bungling opportunities for bringing the war to a conclusion. Eight days later it was reported that he had suggested abandoning some areas of South Vietnam to the Viet Cong. On February 17, 1968, it was reported that McCarthy's campaign had raised only a quarter of the funds it had hoped to raise nationally.

As his volunteers (led by youth coordinator Sam Brown) went door to door in New Hampshire, and as the media began paying more serious attention to the senator, McCarthy began to rise in the polls. When he received 42% of the vote to Johnson's 49% in the March 12 New Hampshire primary (and 20 of New Hampshire's 24 delegates to the Democratic convention), it became clear that there was deep division among Democrats about the war. By this time, Johnson had become inextricably defined by Vietnam, and this demonstration of divided support within his party meant his reelection (only four years after winning the highest percentage of the popular vote in modern history) seemed unlikely. The folk trio Peter, Paul and Mary released a record "Eugene McCarthy For President (If You Love Your Country)", endorsing McCarthy, who they said had stood alone against Johnson over "more timid men" now echoing him.

Kennedy enters the race
On March 16, Robert F. Kennedy announced that he would run; many Democrats saw Kennedy as a stronger candidate than McCarthy. On March 31, Johnson surprised the world by announcing that he would not seek reelection. After that, McCarthy won in Wisconsin, where the Kennedy campaign was still getting organized. McCarthy also won in Oregon against a well-organized Kennedy effort; it was considered his first official victory over Kennedy.

McCarthy styled himself as a clean politician, but criticized his opponents. Known for his wit, when asked if Michigan Governor George Romney's comment that Romney had been "brainwashed" about the Vietnam War had ended Romney's presidential hopes, McCarthy remarked, "Well... no, not really. Anyway, I think in that case a light rinse would have been sufficient." He mocked Kennedy and his supporters. A major gaffe occurred in Oregon, when McCarthy called Kennedy supporters "less intelligent" than his own and belittled Indiana (which had by then gone for Kennedy) for lacking a poet of the stature of Robert Lowell—a friend of McCarthy's who often traveled with him. In May, Kennedy attacked McCarthy's civil rights record.

Some of those who joined McCarthy's effort early on were Kennedy loyalists. Now that Kennedy was in the race, many of them jumped ship, urging McCarthy to drop out and support Kennedy. McCarthy resented that Kennedy had let him do the "dirty work" of challenging Johnson and entered the race only when it became apparent that Johnson was vulnerable. As a result, while he initially entered the campaign with few illusions of winning, McCarthy now devoted himself to beating Kennedy (and Humphrey, who entered the race after Johnson withdrew) and gaining the nomination.

Humphrey, long a champion of labor unions and of civil rights, entered the race with the support of the party "establishment", including most members of Congress, mayors, governors and labor union leaders. He entered too late to compete in any primaries, but had the support of Johnson and many Democratic insiders.

Kennedy, like his brother John in 1960, planned to win the nomination through popular support in the primaries. McCarthy and Kennedy squared off in California, knowing that the result there would be decisive. They both campaigned vigorously up and down the state, with many polls showing them neck-and-neck, and a few predicting a McCarthy victory.

A televised debate between them began to tilt undecided voters away from McCarthy. He made two statements many found ill-considered: that he would accept a government including Communists in South Vietnam, and that only the relocation of inner-city blacks would solve the urban problem. Kennedy pounced, portraying the former idea as soft on communism and the latter as a scheme to bus tens of thousands of ghetto residents into white, conservative Orange County. Kennedy won the California primary on June 4, but was shot after his victory speech at the Ambassador Hotel in Los Angeles, and died soon afterwards. In response, McCarthy refrained from political action for several days. One aide recalled McCarthy sneering about his fallen rival, saying that Kennedy was "demagoguing to the last". Another heard McCarthy say that Kennedy had "brought it on himself"—implying that he had provoked Sirhan Sirhan, the Palestinian gunman convicted of killing him, by promising military support to the state of Israel.

Despite strong showings in several primaries—he won more votes than any other Democratic candidate—McCarthy garnered only 23% of the delegates at the 1968 Democratic National Convention, largely due to the control of state-party organizations over the delegate-selection process. After Kennedy's assassination, many Kennedy delegates, remembering his bitter war of words with McCarthy, chose to support George McGovern rather than McCarthy. Moreover, although Humphrey was not clearly an antiwar candidate, some antiwar Democrats hoped that as president he might succeed where Johnson had failed and extricate the United States from Vietnam. On June 23, 1968, Hubert Humphrey defeated McCarthy, securing significant delegates in their shared home state of Minnesota. Before election day, McCarthy confirmed that he would personally vote for Humphrey, but said that he would go no further than that, stopping short of endorsing him. Although McCarthy did not win the Democratic nomination, the antiwar "New Party", which ran several candidates for president that year, listed him as its nominee on the ballot in Arizona, where he received 2,751 votes, and in Vermont, gaining 579 votes. He also appeared on the Oregon ballot as the New Party choice. He received 20,721 votes as a write-in candidate in California.

Despite McCarthy's anti-Vietnam War stance, North Vietnam's Communist government had a cynical attitude toward him, largely because the lack of money in his campaign made it highly skeptical of what he could achieve, describing McCarthy as "a second-rate politician with little experience or money" in its analysis of the presidential election published in their Army Newspaper dated August 10, 1968.

1972 presidential campaign
McCarthy returned to politics as a candidate for the Democratic presidential nomination in 1972, but he fared poorly in New Hampshire and Wisconsin and soon dropped out.

Illinois was the only primary in which McCarthy actively participated. He got 38% of the vote to the then leading contender Edmund Muskie's 59%, but the media ignored McCarthy's Illinois campaign.

1976 presidential campaign
After his 1972 campaign, McCarthy left the Democratic Party, and ran as an Independent candidate for president in 1976. During that campaign, he took a libertarian stance on civil liberties, promised to create full employment by shortening the work week, came out in favor of nuclear disarmament, attacked the Internal Revenue Service, and said whom he would nominate to various Cabinet posts if elected. Mainly, however, he battled ballot access laws he deemed too restrictive and encouraged voters to reject the two-party system.

His numerous legal battles during the election, along with a strong grassroots effort in friendly states, allowed him to appear on the ballot in 30 states and eased ballot access for later third-party candidates. His party affiliation was variously listed on ballots as "Independent," "McCarthy '76," "Non-Partisan," "Nom. Petition," "Nomination," "Not Designated," and "Court Order". Although he was not on the California and Wyoming ballots, he was recognized as a write-in candidate in those states. In many states, he did not run with a vice-presidential nominee, but he came to have a total of 15 running mates in states where he was required to have one. At least eight of his running mates were women.

Nationally, McCarthy received 740,460 votes, 0.91% of the total, finishing third in the election. His best showing came in Oregon, where he received 40,207 votes, 3.90% of the vote.

Further activism
McCarthy opposed Watergate-era campaign finance laws, becoming a plaintiff in the landmark case Buckley v. Valeo, 424 U.S. 1 (1976), in which the U.S. Supreme Court held that certain provisions of federal campaign finance laws were unconstitutional. McCarthy, the New York Civil Liberties Union, philanthropist Stewart Mott, the Conservative Party of New York State, the Mississippi Republican Party, and the Libertarian Party were the plaintiffs in Buckley, becoming key players in killing campaign spending limits and public financing of political campaigns.

In 1980, dismayed by what he saw as the abject failure of Jimmy Carter's presidency (he later said that "he was the worst president we ever had"), he appeared in a campaign ad for Libertarian candidate Ed Clark and wrote the introduction to Clark's campaign book. He eventually endorsed Ronald Reagan for president.

Final campaigns
In 1982, McCarthy ran for the U.S. Senate in Minnesota but lost the Democratic primary to businessman Mark Dayton, 69% to 24%.

In the 1988 election, McCarthy appeared on the ballot as the presidential candidate of a handful of left-wing state parties, specifically the Consumer parties in Pennsylvania and New Jersey and the Minnesota Progressive Party in Minnesota. In his campaign, he supported trade protectionism, Reagan's Strategic Defense Initiative and the abolition of the two-party system. He received 30,905 votes.

In 1992, returning to the Democratic Party, he entered the New Hampshire presidential primary and campaigned for the Democratic nomination, but was excluded from the first televised debate. Along with other candidates who had been excluded from the 1992 Democratic debates (including two-time New Alliance Party presidential candidate Lenora Fulani, former Irvine, California mayor Larry Agran, Billy Jack actor Tom Laughlin, and others), McCarthy staged protests and took unsuccessful legal action in an attempt to be included in the debates. Unlike the other excluded candidates, McCarthy was a longstanding national figure and had mounted credible campaigns for president in previous elections. He won 108,679 votes in the 1992 primaries. In his campaign for the Democratic nomination, McCarthy proposed the use of import fees to help Japan and Western Europe pay for military security and raise taxes on the wealthy in order to eliminate the national debt.

Publishing
After leaving the Senate in 1971, McCarthy became a senior editor at Harcourt Brace Jovanovich Publishing and a syndicated newspaper columnist. In the 1960s he began writing poetry, and his increased political prominence led to increased interest in his work. "If any of you are secret poets, the best way to break into print is to run for the presidency", he wrote in 1968. He published a collection of poetry in 1997, Cool Reflections: Poetry For The Who, What, When, Where and Especially Why of It All (ISBN 1-57553-595-5).

Personal life
McCarthy and his wife, Abigail Quigley McCarthy, had five children, Christopher Joseph (April 30, 1946 – April 30, 1946), Ellen Anne, Mary Abigail (April 29, 1949 – July 28, 1990), Michael Benet, and Margaret Alice.

In 1969, McCarthy separated from his wife after 24 years of marriage, but the two never divorced. The children stayed with their mother after the separation. According to McCarthy biographer Dominic Sandbrook, McCarthy was involved in a romantic relationship with CBS News correspondent Marya McLaughlin that lasted until McLaughlin's death in 1998.

Death and legacy
McCarthy died of complications from Parkinson's disease at age 89 on December 10, 2005, in a retirement home in Georgetown, Washington, D.C., where he had lived for the previous few years. Former President Bill Clinton gave his eulogy.

Following his death the College of Saint Benedict and Saint John's University renamed their Public Policy Center the Eugene J. McCarthy Center for Public Policy. The Democratic party memorialized his death during the 2008 Democratic National Convention in Denver, Colorado, on August 28, 2008. The memorial included pictures of several prominent Democrats who had died during the four-year period since the 2004 Convention displayed on a large screen. During McCarthy's tribute, the screen displaying his photograph mistakenly left off his first name but included his middle name, calling him "Senator Joseph McCarthy"; Joseph McCarthy was a notable Republican Senator from Wisconsin famous for his anti-Communist campaigning and sparring with journalist Edward R. Murrow.

In 2009, his alma mater, St. John's University, honored McCarthy by establishing the Eugene McCarthy Distinguished Public Service Award.

McCarthy's files as U.S. congressman (Democratic Farmer-Labor) from Minnesota's 4th district (1949–1959) and as U.S. senator from Minnesota (1959–1971) are available at the Minnesota History Center for research. They include executive files, general files, legislative files, personal files, political and campaign (including senatorial, vice presidential, and presidential) files, public relations files, sound and visual materials (with photographs), and speeches.

Books by Eugene McCarthy

 * Frontiers in American Democracy (1960)
 * Dictionary of American Politics (1962)
 * A Liberal Answer to the Conservative Challenge (1964)
 * The Limits of Power: America's Role in the World (1967)
 * The Year of the People (1969)
 * Mr. Raccoon and His Friends (1977; Academy Press Ltd., Chicago, IL); children's stories, illustrated by James Ecklund
 * A Political Bestiary, by Eugene J. McCarthy and James J. Kilpatrick (1979) ISBN 0-380-46508-6
 * The Ultimate Tyranny: The Majority Over the Majority (1980) ISBN 0-15-192581-X
 * Gene McCarthy's Minnesota: Memories of a Native Son (1982) ISBN 0-86683-681-0
 * Complexities and Contrarities (1982) ISBN 0-15-121202-3
 * Up Til Now: A Memoir (1987)
 * Required Reading: A Decade of Political Wit and Wisdom (1988) ISBN 0-15-176880-3
 * Nonfinancial Economics: The Case for Shorter Hours of Work, by Eugene McCarthy and William McGaughey (1989) ISBN 0-275-92514-5
 * A Colony of the World: The United States Today (1992) ISBN 0-7818-0102-8
 * Eugene J. McCarthy: Selected Poems by Eugene J. McCarthy, Ray Howe (1997) ISBN 1-883477-15-8
 * No-Fault Politics (1998) ISBN 0-8129-3016-9
 * 1968: War and Democracy (2000) ISBN 1-883477-37-9
 * Hard Years: Antidotes to Authoritarians (2001) ISBN 1-883477-38-7
 * From Rappahannock County (2002) ISBN 1-883477-51-4
 * Parting Shots from My Brittle Bow: Reflections on American Politics and Life (2005) ISBN 1-55591-528-0

title: Eugene McCarthy